Primitive. The name Eps_i is a term of type (set → prop) → set.

Axiom. (Eps_i_ax) We take the following as an axiom:

∀P : set → prop, ∀x : set, P x → P (Eps_i P)

Definition. We define True to be ∀p : prop, p → p of type prop.

Definition. We define False to be ∀p : prop, p of type prop.

Definition. We define not to be λA : prop ⇒ A → False of type prop → prop.

Notation. We use ¬ as a prefix operator with priority 700 corresponding to applying term not.

Definition. We define and to be λA B : prop ⇒ ∀p : prop, (A → B → p) → p of type prop → prop → prop.

Notation. We use ∧ as an infix operator with priority 780 and which associates to the left corresponding to applying term and.

Definition. We define or to be λA B : prop ⇒ ∀p : prop, (A → p) → (B → p) → p of type prop → prop → prop.

Notation. We use ∨ as an infix operator with priority 785 and which associates to the left corresponding to applying term or.

Definition. We define iff to be λA B : prop ⇒ and (A → B) (B → A) of type prop → prop → prop.

Notation. We use ↔ as an infix operator with priority 805 and no associativity corresponding to applying term iff.

Beginning of Section Eq

Variable A : SType

Definition. We define eq to be λx y : A ⇒ ∀Q : A → A → prop, Q x y → Q y x of type A → A → prop.

Definition. We define neq to be λx y : A ⇒ ¬ eq x y of type A → A → prop.

End of Section Eq

Notation. We use = as an infix operator with priority 502 and no associativity corresponding to applying term eq.

Notation. We use ≠ as an infix operator with priority 502 and no associativity corresponding to applying term neq.

Beginning of Section FE

Variable A B : SType

Axiom. (func_ext) We take the following as an axiom:

∀f g : A → B, (∀x : A, f x = g x) → f = g

End of Section FE

Beginning of Section Ex

Variable A : SType

Definition. We define ex to be λQ : A → prop ⇒ ∀P : prop, (∀x : A, Q x → P) → P of type (A → prop) → prop.

End of Section Ex

Notation. We use ∃ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using ex.

Axiom. (prop_ext) We take the following as an axiom:

∀p q : prop, iff p q → p = q

Primitive. The name In is a term of type set → set → prop.

Notation. We use ∈ as an infix operator with priority 500 and no associativity corresponding to applying term In. Furthermore, we may write ∀ x ∈ A, B to mean ∀ x : set, x ∈ A → B.

Definition. We define Subq to be λA B ⇒ ∀x ∈ A, x ∈ B of type set → set → prop.

Notation. We use ⊆ as an infix operator with priority 500 and no associativity corresponding to applying term Subq. Furthermore, we may write ∀ x ⊆ A, B to mean ∀ x : set, x ⊆ A → B.

Axiom. (set_ext) We take the following as an axiom:

∀X Y : set, X ⊆ Y → Y ⊆ X → X = Y

Axiom. (In_ind) We take the following as an axiom:

∀P : set → prop, (∀X : set, (∀x ∈ X, P x) → P X) → ∀X : set, P X

Notation. We use ∃ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using ex and handling ∈ or ⊆ ascriptions using and.

Primitive. The name Empty is a term of type set.

Axiom. (EmptyAx) We take the following as an axiom:

¬ ∃x : set, x ∈ Empty

Primitive. The name ⋃ is a term of type set → set.

Axiom. (UnionEq) We take the following as an axiom:

∀X x, x ∈ ⋃ X ↔ ∃Y, x ∈ Y ∧ Y ∈ X

Primitive. The name 𝒫 is a term of type set → set.

Axiom. (PowerEq) We take the following as an axiom:

∀X Y : set, Y ∈ 𝒫 X ↔ Y ⊆ X

Primitive. The name Repl is a term of type set → (set → set) → set.

Notation. {B| x ∈ A} is notation for Repl A (λ x . B).

Axiom. (ReplEq) We take the following as an axiom:

∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} ↔ ∃x ∈ A, y = F x

Definition. We define TransSet to be λU : set ⇒ ∀x ∈ U, x ⊆ U of type set → prop.

Definition. We define Union_closed to be λU : set ⇒ ∀X : set, X ∈ U → ⋃ X ∈ U of type set → prop.

Definition. We define Power_closed to be λU : set ⇒ ∀X : set, X ∈ U → 𝒫 X ∈ U of type set → prop.

Definition. We define Repl_closed to be λU : set ⇒ ∀X : set, X ∈ U → ∀F : set → set, (∀x : set, x ∈ X → F x ∈ U) → {F x|x ∈ X} ∈ U of type set → prop.

Definition. We define ZF_closed to be λU : set ⇒ Union_closed U ∧ Power_closed U ∧ Repl_closed U of type set → prop.

Primitive. The name UnivOf is a term of type set → set.

Axiom. (UnivOf_In) We take the following as an axiom:

∀N : set, N ∈ UnivOf N

Axiom. (UnivOf_TransSet) We take the following as an axiom:

∀N : set, TransSet (UnivOf N)

Axiom. (UnivOf_ZF_closed) We take the following as an axiom:

∀N : set, ZF_closed (UnivOf N)

Axiom. (UnivOf_Min) We take the following as an axiom:

∀N U : set, N ∈ U → TransSet U → ZF_closed U → UnivOf N ⊆ U

Axiom. (FalseE) We take the following as an axiom:

False → ∀p : prop, p

Axiom. (TrueI) We take the following as an axiom:

True

Axiom. (notI) We take the following as an axiom:

∀A : prop, (A → False) → ¬ A

Axiom. (notE) We take the following as an axiom:

∀A : prop, ¬ A → A → False

Axiom. (andI) We take the following as an axiom:

∀A B : prop, A → B → A ∧ B

Axiom. (andEL) We take the following as an axiom:

∀A B : prop, A ∧ B → A

Axiom. (andER) We take the following as an axiom:

∀A B : prop, A ∧ B → B

Axiom. (orIL) We take the following as an axiom:

∀A B : prop, A → A ∨ B

Axiom. (orIR) We take the following as an axiom:

∀A B : prop, B → A ∨ B

Axiom. (orE) We take the following as an axiom:

∀A B C : prop, (A → C) → (B → C) → A ∨ B → C

Beginning of Section PropN

Variable P1 P2 P3 : prop

Axiom. (and3I) We take the following as an axiom:

P1 → P2 → P3 → P1 ∧ P2 ∧ P3

Axiom. (and3E) We take the following as an axiom:

P1 ∧ P2 ∧ P3 → (∀p : prop, (P1 → P2 → P3 → p) → p)

Axiom. (or3I1) We take the following as an axiom:

P1 → P1 ∨ P2 ∨ P3

Axiom. (or3I2) We take the following as an axiom:

P2 → P1 ∨ P2 ∨ P3

Axiom. (or3I3) We take the following as an axiom:

P3 → P1 ∨ P2 ∨ P3

Axiom. (or3E) We take the following as an axiom:

P1 ∨ P2 ∨ P3 → (∀p : prop, (P1 → p) → (P2 → p) → (P3 → p) → p)

Variable P4 : prop

Axiom. (and4I) We take the following as an axiom:

P1 → P2 → P3 → P4 → P1 ∧ P2 ∧ P3 ∧ P4

Axiom. (and4E) We take the following as an axiom:

P1 ∧ P2 ∧ P3 ∧ P4 → (∀p : prop, (P1 → P2 → P3 → P4 → p) → p)

Axiom. (or4I1) We take the following as an axiom:

P1 → P1 ∨ P2 ∨ P3 ∨ P4

Axiom. (or4I2) We take the following as an axiom:

P2 → P1 ∨ P2 ∨ P3 ∨ P4

Axiom. (or4I3) We take the following as an axiom:

P3 → P1 ∨ P2 ∨ P3 ∨ P4

Axiom. (or4I4) We take the following as an axiom:

P4 → P1 ∨ P2 ∨ P3 ∨ P4

Axiom. (or4E) We take the following as an axiom:

P1 ∨ P2 ∨ P3 ∨ P4 → (∀p : prop, (P1 → p) → (P2 → p) → (P3 → p) → (P4 → p) → p)

Variable P5 : prop

Axiom. (and5I) We take the following as an axiom:

P1 → P2 → P3 → P4 → P5 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5

Axiom. (and5E) We take the following as an axiom:

P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 → (∀p : prop, (P1 → P2 → P3 → P4 → P5 → p) → p)

Axiom. (or5I1) We take the following as an axiom:

P1 → P1 ∨ P2 ∨ P3 ∨ P4 ∨ P5

Axiom. (or5I2) We take the following as an axiom:

P2 → P1 ∨ P2 ∨ P3 ∨ P4 ∨ P5

Axiom. (or5I3) We take the following as an axiom:

P3 → P1 ∨ P2 ∨ P3 ∨ P4 ∨ P5

Axiom. (or5I4) We take the following as an axiom:

P4 → P1 ∨ P2 ∨ P3 ∨ P4 ∨ P5

Axiom. (or5I5) We take the following as an axiom:

P5 → P1 ∨ P2 ∨ P3 ∨ P4 ∨ P5

Axiom. (or5E) We take the following as an axiom:

P1 ∨ P2 ∨ P3 ∨ P4 ∨ P5 → (∀p : prop, (P1 → p) → (P2 → p) → (P3 → p) → (P4 → p) → (P5 → p) → p)

Variable P6 : prop

Axiom. (and6I) We take the following as an axiom:

P1 → P2 → P3 → P4 → P5 → P6 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6

Axiom. (and6E) We take the following as an axiom:

P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 → (∀p : prop, (P1 → P2 → P3 → P4 → P5 → P6 → p) → p)

Variable P7 : prop

Axiom. (and7I) We take the following as an axiom:

P1 → P2 → P3 → P4 → P5 → P6 → P7 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 ∧ P7

Axiom. (and7E) We take the following as an axiom:

P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 ∧ P7 → (∀p : prop, (P1 → P2 → P3 → P4 → P5 → P6 → P7 → p) → p)

End of Section PropN

Axiom. (iffI) We take the following as an axiom:

∀A B : prop, (A → B) → (B → A) → (A ↔ B)

Axiom. (iffEL) We take the following as an axiom:

∀A B : prop, (A ↔ B) → A → B

Axiom. (iffER) We take the following as an axiom:

∀A B : prop, (A ↔ B) → B → A

Axiom. (iff_ref) We take the following as an axiom:

∀A : prop, A ↔ A

Axiom. (neq_i_sym) We take the following as an axiom:

∀x y, x ≠ y → y ≠ x

Definition. We define nIn to be λx X ⇒ ¬ In x X of type set → set → prop.

Notation. We use ∉ as an infix operator with priority 502 and no associativity corresponding to applying term nIn.

Axiom. (Eps_i_ex) We take the following as an axiom:

∀P : set → prop, (∃x, P x) → P (Eps_i P)

Axiom. (pred_ext) We take the following as an axiom:

∀P Q : set → prop, (∀x, P x ↔ Q x) → P = Q

Axiom. (prop_ext_2) We take the following as an axiom:

∀p q : prop, (p → q) → (q → p) → p = q

Axiom. (pred_ext_2) We take the following as an axiom:

∀P Q : set → prop, P ⊆ Q → Q ⊆ P → P = Q

Axiom. (Subq_ref) We take the following as an axiom:

∀X : set, X ⊆ X

Axiom. (Subq_tra) We take the following as an axiom:

∀X Y Z : set, X ⊆ Y → Y ⊆ Z → X ⊆ Z

Axiom. (Subq_contra) We take the following as an axiom:

∀X Y z : set, X ⊆ Y → z ∉ Y → z ∉ X

Axiom. (EmptyE) We take the following as an axiom:

∀x : set, x ∉ Empty

Axiom. (Subq_Empty) We take the following as an axiom:

∀X : set, Empty ⊆ X

Axiom. (Empty_Subq_eq) We take the following as an axiom:

∀X : set, X ⊆ Empty → X = Empty

Axiom. (Empty_eq) We take the following as an axiom:

∀X : set, (∀x, x ∉ X) → X = Empty

Axiom. (UnionI) We take the following as an axiom:

∀X x Y : set, x ∈ Y → Y ∈ X → x ∈ ⋃ X

Axiom. (UnionE) We take the following as an axiom:

∀X x : set, x ∈ ⋃ X → ∃Y : set, x ∈ Y ∧ Y ∈ X

Axiom. (UnionE_impred) We take the following as an axiom:

∀X x : set, x ∈ ⋃ X → ∀p : prop, (∀Y : set, x ∈ Y → Y ∈ X → p) → p

Axiom. (Union_Empty) We take the following as an axiom:

⋃ Empty = Empty

Axiom. (PowerI) We take the following as an axiom:

∀X Y : set, Y ⊆ X → Y ∈ 𝒫 X

Axiom. (PowerE) We take the following as an axiom:

∀X Y : set, Y ∈ 𝒫 X → Y ⊆ X

Axiom. (Power_Subq) We take the following as an axiom:

∀X Y : set, X ⊆ Y → 𝒫 X ⊆ 𝒫 Y

Axiom. (Empty_In_Power) We take the following as an axiom:

∀X : set, Empty ∈ 𝒫 X

Axiom. (Self_In_Power) We take the following as an axiom:

∀X : set, X ∈ 𝒫 X

Axiom. (Union_Power_Subq) We take the following as an axiom:

∀X : set, ⋃ (𝒫 X) ⊆ X

Axiom. (xm) We take the following as an axiom:

∀P : prop, P ∨ ¬ P

Axiom. (dneg) We take the following as an axiom:

∀P : prop, ¬ ¬ P → P

Axiom. (imp_not_or) We take the following as an axiom:

∀p q : prop, (p → q) → ¬ p ∨ q

Axiom. (not_and_or_demorgan) We take the following as an axiom:

∀p q : prop, ¬ (p ∧ q) → ¬ p ∨ ¬ q

Primitive. The name exactly1of2 is a term of type prop → prop → prop.

Axiom. (exactly1of2_I1) We take the following as an axiom:

∀A B : prop, A → ¬ B → exactly1of2 A B

Axiom. (exactly1of2_I2) We take the following as an axiom:

∀A B : prop, ¬ A → B → exactly1of2 A B

Axiom. (exactly1of2_impI1) We take the following as an axiom:

∀A B : prop, (A → ¬ B) → (¬ A → B) → exactly1of2 A B

Axiom. (exactly1of2_impI2) We take the following as an axiom:

∀A B : prop, (B → ¬ A) → (¬ B → A) → exactly1of2 A B

Axiom. (exactly1of2_E) We take the following as an axiom:

∀A B : prop, exactly1of2 A B → ∀p : prop, (A → ¬ B → p) → (¬ A → B → p) → p

Axiom. (exactly1of2_or) We take the following as an axiom:

∀A B : prop, exactly1of2 A B → A ∨ B

Axiom. (exactly1of2_impn12) We take the following as an axiom:

∀A B : prop, exactly1of2 A B → A → ¬ B

Axiom. (exactly1of2_impn21) We take the following as an axiom:

∀A B : prop, exactly1of2 A B → B → ¬ A

Axiom. (exactly1of2_nimp12) We take the following as an axiom:

∀A B : prop, exactly1of2 A B → ¬ A → B

Axiom. (exactly1of2_nimp21) We take the following as an axiom:

∀A B : prop, exactly1of2 A B → ¬ B → A

Primitive. The name exactly1of3 is a term of type prop → prop → prop → prop.

Axiom. (exactly1of3_I1) We take the following as an axiom:

∀A B C : prop, A → ¬ B → ¬ C → exactly1of3 A B C

Axiom. (exactly1of3_I2) We take the following as an axiom:

∀A B C : prop, ¬ A → B → ¬ C → exactly1of3 A B C

Axiom. (exactly1of3_I3) We take the following as an axiom:

∀A B C : prop, ¬ A → ¬ B → C → exactly1of3 A B C

Axiom. (exactly1of3_impI1) We take the following as an axiom:

∀A B C : prop, (A → ¬ B) → (A → ¬ C) → (B → ¬ C) → (¬ A → B ∨ C) → exactly1of3 A B C

Axiom. (exactly1of3_impI2) We take the following as an axiom:

∀A B C : prop, (B → ¬ A) → (B → ¬ C) → (A → ¬ C) → (¬ B → A ∨ C) → exactly1of3 A B C

Axiom. (exactly1of3_impI3) We take the following as an axiom:

∀A B C : prop, (C → ¬ A) → (C → ¬ B) → (A → ¬ B) → (¬ A → B) → exactly1of3 A B C

Axiom. (exactly1of3_E) We take the following as an axiom:

∀A B C : prop, exactly1of3 A B C → ∀p : prop, (A → ¬ B → ¬ C → p) → (¬ A → B → ¬ C → p) → (¬ A → ¬ B → C → p) → p

Axiom. (exactly1of3_or) We take the following as an axiom:

∀A B C : prop, exactly1of3 A B C → A ∨ B ∨ C

Axiom. (exactly1of3_impn12) We take the following as an axiom:

∀A B C : prop, exactly1of3 A B C → A → ¬ B

Axiom. (exactly1of3_impn13) We take the following as an axiom:

∀A B C : prop, exactly1of3 A B C → A → ¬ C

Axiom. (exactly1of3_impn21) We take the following as an axiom:

∀A B C : prop, exactly1of3 A B C → B → ¬ A

Axiom. (exactly1of3_impn23) We take the following as an axiom:

∀A B C : prop, exactly1of3 A B C → B → ¬ C

Axiom. (exactly1of3_impn31) We take the following as an axiom:

∀A B C : prop, exactly1of3 A B C → C → ¬ A

Axiom. (exactly1of3_impn32) We take the following as an axiom:

∀A B C : prop, exactly1of3 A B C → C → ¬ B

Axiom. (exactly1of3_nimp1) We take the following as an axiom:

∀A B C : prop, exactly1of3 A B C → ¬ A → B ∨ C

Axiom. (exactly1of3_nimp2) We take the following as an axiom:

∀A B C : prop, exactly1of3 A B C → ¬ B → A ∨ C

Axiom. (exactly1of3_nimp3) We take the following as an axiom:

∀A B C : prop, exactly1of3 A B C → ¬ C → A ∨ B

Axiom. (ReplI) We take the following as an axiom:

∀A : set, ∀F : set → set, ∀x : set, x ∈ A → F x ∈ {F x|x ∈ A}

Axiom. (ReplE) We take the following as an axiom:

∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} → ∃x ∈ A, y = F x

Axiom. (ReplE_impred) We take the following as an axiom:

∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} → ∀p : prop, (∀x : set, x ∈ A → y = F x → p) → p

Axiom. (Repl_Empty) We take the following as an axiom:

∀F : set → set, {F x|x ∈ Empty} = Empty

Axiom. (ReplEq_ext_sub) We take the following as an axiom:

∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → {F x|x ∈ X} ⊆ {G x|x ∈ X}

Axiom. (ReplEq_ext) We take the following as an axiom:

∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → {F x|x ∈ X} = {G x|x ∈ X}

Primitive. The name If_i is a term of type prop → set → set → set.

Notation. if cond then T else E is notation corresponding to If_i type cond T E where type is the inferred type of T.

Axiom. (If_i_correct) We take the following as an axiom:

∀p : prop, ∀x y : set, p ∧ (if p then x else y) = x ∨ ¬ p ∧ (if p then x else y) = y

Axiom. (If_i_0) We take the following as an axiom:

∀p : prop, ∀x y : set, ¬ p → (if p then x else y) = y

Axiom. (If_i_1) We take the following as an axiom:

∀p : prop, ∀x y : set, p → (if p then x else y) = x

Axiom. (If_i_or) We take the following as an axiom:

∀p : prop, ∀x y : set, (if p then x else y) = x ∨ (if p then x else y) = y

Axiom. (If_i_eta) We take the following as an axiom:

∀p : prop, ∀x : set, (if p then x else x) = x

Primitive. The name UPair is a term of type set → set → set.

Notation. {x,y} is notation for UPair x y.

Axiom. (UPairE) We take the following as an axiom:

∀x y z : set, x ∈ {y,z} → x = y ∨ x = z

Axiom. (UPairI1) We take the following as an axiom:

∀y z : set, y ∈ {y,z}

Axiom. (UPairI2) We take the following as an axiom:

∀y z : set, z ∈ {y,z}

Axiom. (UPair_com) We take the following as an axiom:

∀x y : set, {x,y} = {y,x}

Primitive. The name Sing is a term of type set → set.

Notation. {x} is notation for Sing x.

Axiom. (SingI) We take the following as an axiom:

∀x : set, x ∈ {x}

Axiom. (SingE) We take the following as an axiom:

∀x y : set, y ∈ {x} → y = x

Primitive. The name binunion is a term of type set → set → set.

Notation. We use ∪ as an infix operator with priority 345 and which associates to the left corresponding to applying term binunion.

Axiom. (binunionI1) We take the following as an axiom:

∀X Y z : set, z ∈ X → z ∈ X ∪ Y

Axiom. (binunionI2) We take the following as an axiom:

∀X Y z : set, z ∈ Y → z ∈ X ∪ Y

Axiom. (binunionE) We take the following as an axiom:

∀X Y z : set, z ∈ X ∪ Y → z ∈ X ∨ z ∈ Y

Definition. We define SetAdjoin to be λX y ⇒ X ∪ {y} of type set → set → set.

Notation. We now use the set enumeration notation {...,...,...} in general. If 0 elements are given, then Empty is used to form the corresponding term. If 1 element is given, then Sing is used to form the corresponding term. If 2 elements are given, then UPair is used to form the corresponding term. If more than elements are given, then SetAdjoin is used to reduce to the case with one fewer elements.

Axiom. (Power_0_Sing_0) We take the following as an axiom:

𝒫 Empty = {Empty}

Axiom. (Repl_UPair) We take the following as an axiom:

∀F : set → set, ∀x y : set, {F z|z ∈ {x,y}} = {F x,F y}

Axiom. (Repl_Sing) We take the following as an axiom:

∀F : set → set, ∀x : set, {F z|z ∈ {x}} = {F x}

Axiom. (Repl_restr) We take the following as an axiom:

∀X : set, ∀F G : set → set, (∀x : set, x ∈ X → F x = G x) → {F x|x ∈ X} = {G x|x ∈ X}

Primitive. The name famunion is a term of type set → (set → set) → set.

Notation. We use ⋃ x [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using famunion.

Axiom. (famunionI) We take the following as an axiom:

∀X : set, ∀F : (set → set), ∀x y : set, x ∈ X → y ∈ F x → y ∈ ⋃_{x ∈ X}F x

Axiom. (famunionE) We take the following as an axiom:

∀X : set, ∀F : (set → set), ∀y : set, y ∈ (⋃_{x ∈ X}F x) → ∃x ∈ X, y ∈ F x

Axiom. (famunionE_impred) We take the following as an axiom:

∀X : set, ∀F : (set → set), ∀y : set, y ∈ (⋃_{x ∈ X}F x) → ∀p : prop, (∀x, x ∈ X → y ∈ F x → p) → p

Axiom. (UnionEq_famunionId) We take the following as an axiom:

∀X : set, ⋃ X = ⋃_{x ∈ X}x

Axiom. (ReplEq_famunion_Sing) We take the following as an axiom:

∀X : set, ∀F : (set → set), {F x|x ∈ X} = ⋃_{x ∈ X}{F x}

Axiom. (Power_Sing) We take the following as an axiom:

∀x : set, 𝒫 {x} = {Empty,{x}}

Axiom. (Power_Sing_0) We take the following as an axiom:

𝒫 {Empty} = {Empty,{Empty}}

Primitive. The name Sep is a term of type set → (set → prop) → set.

Notation. {x ∈ A | B} is notation for Sep A (λ x . B).

Axiom. (SepI) We take the following as an axiom:

∀X : set, ∀P : (set → prop), ∀x : set, x ∈ X → P x → x ∈ {x ∈ X|P x}

Axiom. (SepE) We take the following as an axiom:

∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X ∧ P x

Axiom. (SepE1) We take the following as an axiom:

∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X

Axiom. (SepE2) We take the following as an axiom:

∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → P x

Axiom. (Sep_Subq) We take the following as an axiom:

∀X : set, ∀P : set → prop, {x ∈ X|P x} ⊆ X

Axiom. (Sep_In_Power) We take the following as an axiom:

∀X : set, ∀P : set → prop, {x ∈ X|P x} ∈ 𝒫 X

Primitive. The name ReplSep is a term of type set → (set → prop) → (set → set) → set.

Notation. {B| x ∈ A, C} is notation for ReplSep A (λ x . C) (λ x . B).

Axiom. (ReplSepI) We take the following as an axiom:

∀X : set, ∀P : set → prop, ∀F : set → set, ∀x : set, x ∈ X → P x → F x ∈ {F x|x ∈ X, P x}

Axiom. (ReplSepE) We take the following as an axiom:

∀X : set, ∀P : set → prop, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ X, P x} → ∃x : set, x ∈ X ∧ P x ∧ y = F x

Axiom. (ReplSepE_impred) We take the following as an axiom:

∀X : set, ∀P : set → prop, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ X, P x} → ∀p : prop, (∀x ∈ X, P x → y = F x → p) → p

Primitive. The name ReplSep2 is a term of type set → (set → set) → (set → set → prop) → (set → set → set) → set.

Axiom. (ReplSep2I) We take the following as an axiom:

∀A, ∀B : set → set, ∀P : set → set → prop, ∀F : set → set → set, ∀x ∈ A, ∀y ∈ B x, P x y → F x y ∈ ReplSep2 A B P F

Axiom. (ReplSep2E_impred) We take the following as an axiom:

∀A, ∀B : set → set, ∀P : set → set → prop, ∀F : set → set → set, ∀r ∈ ReplSep2 A B P F, ∀p : prop, (∀x ∈ A, ∀y ∈ B x, P x y → r = F x y → p) → p

Axiom. (ReplSep2E) We take the following as an axiom:

∀A, ∀B : set → set, ∀P : set → set → prop, ∀F : set → set → set, ∀r ∈ ReplSep2 A B P F, ∃x ∈ A, ∃y ∈ B x, P x y ∧ r = F x y

Axiom. (binunion_asso) We take the following as an axiom:

∀X Y Z : set, X ∪ (Y ∪ Z) = (X ∪ Y) ∪ Z

Axiom. (binunion_com) We take the following as an axiom:

∀X Y : set, X ∪ Y = Y ∪ X

Axiom. (binunion_idl) We take the following as an axiom:

∀X : set, Empty ∪ X = X

Axiom. (binunion_idr) We take the following as an axiom:

∀X : set, X ∪ Empty = X

Axiom. (binunion_idem) We take the following as an axiom:

∀X : set, X ∪ X = X

Axiom. (binunion_Subq_1) We take the following as an axiom:

∀X Y : set, X ⊆ X ∪ Y

Axiom. (binunion_Subq_2) We take the following as an axiom:

∀X Y : set, Y ⊆ X ∪ Y

Axiom. (binunion_Subq_min) We take the following as an axiom:

∀X Y Z : set, X ⊆ Z → Y ⊆ Z → X ∪ Y ⊆ Z

Axiom. (Subq_binunion_eq) We take the following as an axiom:

∀X Y, (X ⊆ Y) = (X ∪ Y = Y)

Axiom. (binunion_nIn_I) We take the following as an axiom:

∀X Y z : set, z ∉ X → z ∉ Y → z ∉ X ∪ Y

Axiom. (binunion_nIn_E) We take the following as an axiom:

∀X Y z : set, z ∉ X ∪ Y → z ∉ X ∧ z ∉ Y

Primitive. The name binintersect is a term of type set → set → set.

Notation. We use ∩ as an infix operator with priority 340 and which associates to the left corresponding to applying term binintersect.

Axiom. (binintersectI) We take the following as an axiom:

∀X Y z, z ∈ X → z ∈ Y → z ∈ X ∩ Y

Axiom. (binintersectE) We take the following as an axiom:

∀X Y z, z ∈ X ∩ Y → z ∈ X ∧ z ∈ Y

Axiom. (binintersectE1) We take the following as an axiom:

∀X Y z, z ∈ X ∩ Y → z ∈ X

Axiom. (binintersectE2) We take the following as an axiom:

∀X Y z, z ∈ X ∩ Y → z ∈ Y

Axiom. (binintersect_Subq_1) We take the following as an axiom:

∀X Y : set, X ∩ Y ⊆ X

Axiom. (binintersect_Subq_2) We take the following as an axiom:

∀X Y : set, X ∩ Y ⊆ Y

Axiom. (binintersect_Subq_eq_1) We take the following as an axiom:

∀X Y, X ⊆ Y → X ∩ Y = X

Axiom. (binintersect_Subq_max) We take the following as an axiom:

∀X Y Z : set, Z ⊆ X → Z ⊆ Y → Z ⊆ X ∩ Y

Axiom. (binintersect_asso) We take the following as an axiom:

∀X Y Z : set, X ∩ (Y ∩ Z) = (X ∩ Y) ∩ Z

Axiom. (binintersect_com) We take the following as an axiom:

∀X Y : set, X ∩ Y = Y ∩ X

Axiom. (binintersect_annil) We take the following as an axiom:

∀X : set, Empty ∩ X = Empty

Axiom. (binintersect_annir) We take the following as an axiom:

∀X : set, X ∩ Empty = Empty

Axiom. (binintersect_idem) We take the following as an axiom:

∀X : set, X ∩ X = X

Axiom. (binintersect_binunion_distr) We take the following as an axiom:

∀X Y Z : set, X ∩ (Y ∪ Z) = X ∩ Y ∪ X ∩ Z

Axiom. (binunion_binintersect_distr) We take the following as an axiom:

∀X Y Z : set, X ∪ Y ∩ Z = (X ∪ Y) ∩ (X ∪ Z)

Axiom. (Subq_binintersection_eq) We take the following as an axiom:

∀X Y : set, (X ⊆ Y) = (X ∩ Y = X)

Axiom. (binintersect_nIn_I1) We take the following as an axiom:

∀X Y z : set, z ∉ X → z ∉ X ∩ Y

Axiom. (binintersect_nIn_I2) We take the following as an axiom:

∀X Y z : set, z ∉ Y → z ∉ X ∩ Y

Axiom. (binintersect_nIn_E) We take the following as an axiom:

∀X Y z : set, z ∉ X ∩ Y → z ∉ X ∨ z ∉ Y

Primitive. The name setminus is a term of type set → set → set.

Notation. We use ∖ as an infix operator with priority 350 and no associativity corresponding to applying term setminus.

Axiom. (setminusI) We take the following as an axiom:

∀X Y z, (z ∈ X) → (z ∉ Y) → z ∈ X ∖ Y

Axiom. (setminusE) We take the following as an axiom:

∀X Y z, (z ∈ X ∖ Y) → z ∈ X ∧ z ∉ Y

Axiom. (setminusE1) We take the following as an axiom:

∀X Y z, (z ∈ X ∖ Y) → z ∈ X

Axiom. (setminusE2) We take the following as an axiom:

∀X Y z, (z ∈ X ∖ Y) → z ∉ Y

Axiom. (setminus_Subq) We take the following as an axiom:

∀X Y : set, X ∖ Y ⊆ X

Axiom. (setminus_Subq_contra) We take the following as an axiom:

∀X Y Z : set, Z ⊆ Y → X ∖ Y ⊆ X ∖ Z

Axiom. (setminus_nIn_I1) We take the following as an axiom:

∀X Y z, z ∉ X → z ∉ X ∖ Y

Axiom. (setminus_nIn_I2) We take the following as an axiom:

∀X Y z, z ∈ Y → z ∉ X ∖ Y

Axiom. (setminus_nIn_E) We take the following as an axiom:

∀X Y z, z ∉ X ∖ Y → z ∉ X ∨ z ∈ Y

Axiom. (setminus_selfannih) We take the following as an axiom:

∀X : set, (X ∖ X) = Empty

Axiom. (setminus_binintersect) We take the following as an axiom:

∀X Y Z : set, X ∖ Y ∩ Z = (X ∖ Y) ∪ (X ∖ Z)

Axiom. (setminus_binunion) We take the following as an axiom:

∀X Y Z : set, X ∖ Y ∪ Z = (X ∖ Y) ∖ Z

Axiom. (binintersect_setminus) We take the following as an axiom:

∀X Y Z : set, (X ∩ Y) ∖ Z = X ∩ (Y ∖ Z)

Axiom. (binunion_setminus) We take the following as an axiom:

∀X Y Z : set, X ∪ Y ∖ Z = (X ∖ Z) ∪ (Y ∖ Z)

Axiom. (setminus_setminus) We take the following as an axiom:

∀X Y Z : set, X ∖ (Y ∖ Z) = (X ∖ Y) ∪ (X ∩ Z)

Axiom. (setminus_annil) We take the following as an axiom:

∀X : set, Empty ∖ X = Empty

Axiom. (setminus_idr) We take the following as an axiom:

∀X : set, X ∖ Empty = X

Axiom. (In_irref) We take the following as an axiom:

∀x, x ∉ x

Axiom. (In_no2cycle) We take the following as an axiom:

∀x y, x ∈ y → y ∈ x → False

Axiom. (In_no3cycle) We take the following as an axiom:

∀x y z, x ∈ y → y ∈ z → z ∈ x → False

Primitive. The name ordsucc is a term of type set → set.

Axiom. (ordsuccI1) We take the following as an axiom:

∀x : set, x ⊆ ordsucc x

Axiom. (ordsuccI2) We take the following as an axiom:

∀x : set, x ∈ ordsucc x

Axiom. (ordsuccE) We take the following as an axiom:

∀x y : set, y ∈ ordsucc x → y ∈ x ∨ y = x

Notation. Natural numbers 0,1,2,... are notation for the terms formed using Empty as 0 and forming successors with ordsucc.

Axiom. (neq_0_ordsucc) We take the following as an axiom:

∀a : set, 0 ≠ ordsucc a

Axiom. (neq_ordsucc_0) We take the following as an axiom:

∀a : set, ordsucc a ≠ 0

Axiom. (ordsucc_inj) We take the following as an axiom:

∀a b : set, ordsucc a = ordsucc b → a = b

Axiom. (ordsucc_inj_contra) We take the following as an axiom:

∀a b : set, a ≠ b → ordsucc a ≠ ordsucc b

Axiom. (In_0_1) We take the following as an axiom:

0 ∈ 1

Axiom. (In_0_2) We take the following as an axiom:

0 ∈ 2

Axiom. (In_1_2) We take the following as an axiom:

1 ∈ 2

Definition. We define nat_p to be λn : set ⇒ ∀p : set → prop, p 0 → (∀x : set, p x → p (ordsucc x)) → p n of type set → prop.

Axiom. (nat_0) We take the following as an axiom:

nat_p 0

Axiom. (nat_ordsucc) We take the following as an axiom:

∀n : set, nat_p n → nat_p (ordsucc n)

Axiom. (nat_1) We take the following as an axiom:

nat_p 1

Axiom. (nat_2) We take the following as an axiom:

nat_p 2

Axiom. (nat_3) We take the following as an axiom:

nat_p 3

Axiom. (nat_4) We take the following as an axiom:

nat_p 4

Axiom. (nat_5) We take the following as an axiom:

nat_p 5

Axiom. (nat_6) We take the following as an axiom:

nat_p 6

Axiom. (nat_0_in_ordsucc) We take the following as an axiom:

∀n, nat_p n → 0 ∈ ordsucc n

Axiom. (nat_ordsucc_in_ordsucc) We take the following as an axiom:

∀n, nat_p n → ∀m ∈ n, ordsucc m ∈ ordsucc n

Axiom. (nat_ind) We take the following as an axiom:

∀p : set → prop, p 0 → (∀n, nat_p n → p n → p (ordsucc n)) → ∀n, nat_p n → p n

Axiom. (nat_inv) We take the following as an axiom:

∀n, nat_p n → n = 0 ∨ ∃x, nat_p x ∧ n = ordsucc x

Axiom. (nat_complete_ind) We take the following as an axiom:

∀p : set → prop, (∀n, nat_p n → (∀m ∈ n, p m) → p n) → ∀n, nat_p n → p n

Axiom. (nat_p_trans) We take the following as an axiom:

∀n, nat_p n → ∀m ∈ n, nat_p m

Axiom. (nat_trans) We take the following as an axiom:

∀n, nat_p n → ∀m ∈ n, m ⊆ n

Axiom. (nat_ordsucc_trans) We take the following as an axiom:

∀n, nat_p n → ∀m ∈ ordsucc n, m ⊆ n

Axiom. (Union_ordsucc_eq) We take the following as an axiom:

∀n, nat_p n → ⋃ (ordsucc n) = n

Axiom. (In_0_3) We take the following as an axiom:

0 ∈ 3

Axiom. (In_1_3) We take the following as an axiom:

1 ∈ 3

Axiom. (In_2_3) We take the following as an axiom:

2 ∈ 3

Axiom. (In_0_4) We take the following as an axiom:

0 ∈ 4

Axiom. (In_1_4) We take the following as an axiom:

1 ∈ 4

Axiom. (In_2_4) We take the following as an axiom:

2 ∈ 4

Axiom. (In_3_4) We take the following as an axiom:

3 ∈ 4

Axiom. (In_0_5) We take the following as an axiom:

0 ∈ 5

Axiom. (In_1_5) We take the following as an axiom:

1 ∈ 5

Axiom. (In_2_5) We take the following as an axiom:

2 ∈ 5

Axiom. (In_3_5) We take the following as an axiom:

3 ∈ 5

Axiom. (In_4_5) We take the following as an axiom:

4 ∈ 5

Axiom. (In_0_6) We take the following as an axiom:

0 ∈ 6

Axiom. (In_1_6) We take the following as an axiom:

1 ∈ 6

Axiom. (In_2_6) We take the following as an axiom:

2 ∈ 6

Axiom. (In_3_6) We take the following as an axiom:

3 ∈ 6

Axiom. (In_4_6) We take the following as an axiom:

4 ∈ 6

Axiom. (In_5_6) We take the following as an axiom:

5 ∈ 6

Axiom. (cases_1) We take the following as an axiom:

∀i ∈ 1, ∀p : set → prop, p 0 → p i

Axiom. (cases_2) We take the following as an axiom:

∀i ∈ 2, ∀p : set → prop, p 0 → p 1 → p i

Axiom. (cases_3) We take the following as an axiom:

∀i ∈ 3, ∀p : set → prop, p 0 → p 1 → p 2 → p i

Axiom. (cases_4) We take the following as an axiom:

∀i ∈ 4, ∀p : set → prop, p 0 → p 1 → p 2 → p 3 → p i

Axiom. (cases_5) We take the following as an axiom:

∀i ∈ 5, ∀p : set → prop, p 0 → p 1 → p 2 → p 3 → p 4 → p i

Axiom. (cases_6) We take the following as an axiom:

∀i ∈ 6, ∀p : set → prop, p 0 → p 1 → p 2 → p 3 → p 4 → p 5 → p i

Axiom. (neq_0_1) We take the following as an axiom:

0 ≠ 1

Axiom. (neq_0_2) We take the following as an axiom:

0 ≠ 2

Axiom. (neq_1_2) We take the following as an axiom:

1 ≠ 2

Axiom. (neq_1_0) We take the following as an axiom:

1 ≠ 0

Axiom. (neq_2_0) We take the following as an axiom:

2 ≠ 0

Axiom. (neq_2_1) We take the following as an axiom:

2 ≠ 1

Axiom. (neq_3_0) We take the following as an axiom:

3 ≠ 0

Axiom. (neq_3_1) We take the following as an axiom:

3 ≠ 1

Axiom. (neq_3_2) We take the following as an axiom:

3 ≠ 2

Axiom. (neq_4_0) We take the following as an axiom:

4 ≠ 0

Axiom. (neq_4_1) We take the following as an axiom:

4 ≠ 1

Axiom. (neq_4_2) We take the following as an axiom:

4 ≠ 2

Axiom. (neq_4_3) We take the following as an axiom:

4 ≠ 3

Axiom. (neq_5_0) We take the following as an axiom:

5 ≠ 0

Axiom. (neq_5_1) We take the following as an axiom:

5 ≠ 1

Axiom. (neq_5_2) We take the following as an axiom:

5 ≠ 2

Axiom. (neq_5_3) We take the following as an axiom:

5 ≠ 3

Axiom. (neq_5_4) We take the following as an axiom:

5 ≠ 4

Axiom. (ZF_closed_I) We take the following as an axiom:

∀U, Union_closed U → Power_closed U → Repl_closed U → ZF_closed U

Axiom. (ZF_closed_E) We take the following as an axiom:

∀U, ZF_closed U → ∀p : prop, (Union_closed U → Power_closed U → Repl_closed U → p) → p

Axiom. (ZF_Union_closed) We take the following as an axiom:

∀U, ZF_closed U → ∀X ∈ U, ⋃ X ∈ U

Axiom. (ZF_Power_closed) We take the following as an axiom:

∀U, ZF_closed U → ∀X ∈ U, 𝒫 X ∈ U

Axiom. (ZF_Repl_closed) We take the following as an axiom:

∀U, ZF_closed U → ∀X ∈ U, ∀F : set → set, (∀x ∈ X, F x ∈ U) → {F x|x ∈ X} ∈ U

Axiom. (ZF_UPair_closed) We take the following as an axiom:

∀U, ZF_closed U → ∀x y ∈ U, {x,y} ∈ U

Axiom. (ZF_Sing_closed) We take the following as an axiom:

∀U, ZF_closed U → ∀x ∈ U, {x} ∈ U

Axiom. (ZF_binunion_closed) We take the following as an axiom:

∀U, ZF_closed U → ∀X Y ∈ U, (X ∪ Y) ∈ U

Axiom. (ZF_ordsucc_closed) We take the following as an axiom:

∀U, ZF_closed U → ∀x ∈ U, ordsucc x ∈ U

Axiom. (nat_p_UnivOf_Empty) We take the following as an axiom:

∀n : set, nat_p n → n ∈ UnivOf Empty

Primitive. The name ω is a term of type set.

Axiom. (omega_nat_p) We take the following as an axiom:

∀n ∈ ω, nat_p n

Axiom. (nat_p_omega) We take the following as an axiom:

∀n : set, nat_p n → n ∈ ω

Axiom. (omega_ordsucc) We take the following as an axiom:

∀n ∈ ω, ordsucc n ∈ ω

Definition. We define ordinal to be λalpha : set ⇒ TransSet alpha ∧ ∀beta ∈ alpha, TransSet beta of type set → prop.

Axiom. (ordinal_TransSet) We take the following as an axiom:

∀alpha : set, ordinal alpha → TransSet alpha

Axiom. (ordinal_In_TransSet) We take the following as an axiom:

∀alpha : set, ordinal alpha → ∀beta ∈ alpha, TransSet beta

Axiom. (ordinal_Empty) We take the following as an axiom:

ordinal Empty

Axiom. (ordinal_Hered) We take the following as an axiom:

∀alpha : set, ordinal alpha → ∀beta ∈ alpha, ordinal beta

Axiom. (TransSet_ordsucc) We take the following as an axiom:

∀X : set, TransSet X → TransSet (ordsucc X)

Axiom. (ordinal_ordsucc) We take the following as an axiom:

∀alpha : set, ordinal alpha → ordinal (ordsucc alpha)

Axiom. (nat_p_ordinal) We take the following as an axiom:

∀n : set, nat_p n → ordinal n

Axiom. (ordinal_1) We take the following as an axiom:

ordinal 1

Axiom. (ordinal_2) We take the following as an axiom:

ordinal 2

Axiom. (omega_TransSet) We take the following as an axiom:

TransSet ω

Axiom. (omega_ordinal) We take the following as an axiom:

ordinal ω

Axiom. (ordsucc_omega_ordinal) We take the following as an axiom:

ordinal (ordsucc ω)

Axiom. (TransSet_ordsucc_In_Subq) We take the following as an axiom:

∀X : set, TransSet X → ∀x ∈ X, ordsucc x ⊆ X

Axiom. (ordinal_ordsucc_In_Subq) We take the following as an axiom:

∀alpha, ordinal alpha → ∀beta ∈ alpha, ordsucc beta ⊆ alpha

Axiom. (ordinal_trichotomy_or) We take the following as an axiom:

∀alpha beta : set, ordinal alpha → ordinal beta → alpha ∈ beta ∨ alpha = beta ∨ beta ∈ alpha

Axiom. (ordinal_In_Or_Subq) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → alpha ∈ beta ∨ beta ⊆ alpha

Axiom. (ordinal_linear) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta ∨ beta ⊆ alpha

Axiom. (ordinal_ordsucc_In_eq) We take the following as an axiom:

∀alpha beta, ordinal alpha → beta ∈ alpha → ordsucc beta ∈ alpha ∨ alpha = ordsucc beta

Axiom. (ordinal_lim_or_succ) We take the following as an axiom:

∀alpha, ordinal alpha → (∀beta ∈ alpha, ordsucc beta ∈ alpha) ∨ (∃beta ∈ alpha, alpha = ordsucc beta)

Axiom. (ordinal_ordsucc_In) We take the following as an axiom:

∀alpha, ordinal alpha → ∀beta ∈ alpha, ordsucc beta ∈ ordsucc alpha

Axiom. (ordinal_Union) We take the following as an axiom:

∀X, (∀x ∈ X, ordinal x) → ordinal (⋃ X)

Axiom. (ordinal_famunion) We take the following as an axiom:

∀X, ∀F : set → set, (∀x ∈ X, ordinal (F x)) → ordinal (⋃_{x ∈ X}F x)

Axiom. (ordinal_binintersect) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → ordinal (alpha ∩ beta)

Axiom. (ordinal_binunion) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → ordinal (alpha ∪ beta)

Axiom. (ordinal_Sep) We take the following as an axiom:

∀alpha, ordinal alpha → ∀p : set → prop, (∀beta ∈ alpha, ∀gamma ∈ beta, p beta → p gamma) → ordinal {beta ∈ alpha|p beta}

Definition. We define inj to be λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀u v ∈ X, f u = f v → u = v) of type set → set → (set → set) → prop.

Definition. We define surj to be λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀w ∈ Y, ∃u ∈ X, f u = w) of type set → set → (set → set) → prop.

Definition. We define bij to be λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀u v ∈ X, f u = f v → u = v) ∧ (∀w ∈ Y, ∃u ∈ X, f u = w) of type set → set → (set → set) → prop.

Axiom. (bijI) We take the following as an axiom:

∀X Y, ∀f : set → set, (∀u ∈ X, f u ∈ Y) → (∀u v ∈ X, f u = f v → u = v) → (∀w ∈ Y, ∃u ∈ X, f u = w) → bij X Y f

Axiom. (bijE) We take the following as an axiom:

∀X Y, ∀f : set → set, bij X Y f → ∀p : prop, ((∀u ∈ X, f u ∈ Y) → (∀u v ∈ X, f u = f v → u = v) → (∀w ∈ Y, ∃u ∈ X, f u = w) → p) → p

Primitive. The name inv is a term of type set → (set → set) → set → set.

Axiom. (surj_rinv) We take the following as an axiom:

∀X Y, ∀f : set → set, (∀w ∈ Y, ∃u ∈ X, f u = w) → ∀y ∈ Y, inv X f y ∈ X ∧ f (inv X f y) = y

Axiom. (inj_linv) We take the following as an axiom:

∀X Y, ∀f : set → set, (∀u v ∈ X, f u = f v → u = v) → ∀x ∈ X, inv X f (f x) = x

Axiom. (bij_inv) We take the following as an axiom:

∀X Y, ∀f : set → set, bij X Y f → bij Y X (inv X f)

Axiom. (bij_comp) We take the following as an axiom:

∀X Y Z : set, ∀f g : set → set, bij X Y f → bij Y Z g → bij X Z (λx ⇒ g (f x))

Axiom. (bij_id) We take the following as an axiom:

∀X, bij X X (λx ⇒ x)

Axiom. (bij_inj) We take the following as an axiom:

∀X Y, ∀f : set → set, bij X Y f → inj X Y f

Axiom. (bij_surj) We take the following as an axiom:

∀X Y, ∀f : set → set, bij X Y f → surj X Y f

Axiom. (inj_surj_bij) We take the following as an axiom:

∀X Y, ∀f : set → set, inj X Y f → surj X Y f → bij X Y f

Axiom. (surj_inv_inj) We take the following as an axiom:

∀X Y, ∀f : set → set, (∀y ∈ Y, ∃x ∈ X, f x = y) → inj Y X (inv X f)

Definition. We define atleastp to be λX Y : set ⇒ ∃f : set → set, inj X Y f of type set → set → prop.

Definition. We define equip to be λX Y : set ⇒ ∃f : set → set, bij X Y f of type set → set → prop.

Axiom. (equip_ref) We take the following as an axiom:

∀X, equip X X

Axiom. (equip_sym) We take the following as an axiom:

∀X Y, equip X Y → equip Y X

Axiom. (equip_tra) We take the following as an axiom:

∀X Y Z, equip X Y → equip Y Z → equip X Z

Definition. We define finite to be λX ⇒ ∃n ∈ ω, equip X n of type set → prop.

Definition. We define infinite to be λX ⇒ ¬ finite X of type set → prop.

Axiom. (KnasterTarski_set) We take the following as an axiom:

∀A, ∀F : set → set, (∀U ∈ 𝒫 A, F U ∈ 𝒫 A) → (∀U V ∈ 𝒫 A, U ⊆ V → F U ⊆ F V) → ∃Y ∈ 𝒫 A, F Y = Y

Axiom. (image_In_Power) We take the following as an axiom:

∀A B, ∀f : set → set, (∀x ∈ A, f x ∈ B) → ∀U ∈ 𝒫 A, {f x|x ∈ U} ∈ 𝒫 B

Axiom. (image_monotone) We take the following as an axiom:

∀f : set → set, ∀U V, U ⊆ V → {f x|x ∈ U} ⊆ {f x|x ∈ V}

Axiom. (setminus_In_Power) We take the following as an axiom:

∀A U, A ∖ U ∈ 𝒫 A

Axiom. (setminus_antimonotone) We take the following as an axiom:

∀A U V, U ⊆ V → A ∖ V ⊆ A ∖ U

Axiom. (SchroederBernstein) We take the following as an axiom:

∀A B, ∀f g : set → set, inj A B f → inj B A g → equip A B

Axiom. (f_eq_i) We take the following as an axiom:

∀f : set → set, ∀x y, x = y → f x = f y

Axiom. (f_eq_i_i) We take the following as an axiom:

∀f : set → set → set, ∀x y z w, x = y → z = w → f x z = f y w

Axiom. (eq_i_tra) We take the following as an axiom:

∀x y z, x = y → y = z → x = z

Definition. We define nSubq to be λX Y ⇒ ¬ Subq X Y of type set → set → prop.

Notation. We use ⊈ as an infix operator with priority 502 and no associativity corresponding to applying term nSubq.

Axiom. (Sing_inv) We take the following as an axiom:

∀x Y, {x} = Y → x ∈ Y ∧ ∀y ∈ Y, y = x

Axiom. (TransSet_In_ordsucc_Subq) We take the following as an axiom:

∀x y, TransSet y → x ∈ ordsucc y → x ⊆ y

Axiom. (inv_Repl_eq) We take the following as an axiom:

∀X, ∀f g : set → set, (∀x ∈ X, f (g x) = x) → {f y|y ∈ {g x|x ∈ X}} = X

Axiom. (invol_Repl_eq) We take the following as an axiom:

∀X, ∀f : set → set, (∀x ∈ X, f (f x) = x) → {f y|y ∈ {f x|x ∈ X}} = X

Axiom. (Eps_i_set_R) We take the following as an axiom:

∀X : set, ∀P : set → prop, ∀x ∈ X, P x → Eps_i (λx ⇒ x ∈ X ∧ P x) ∈ X ∧ P (Eps_i (λx ⇒ x ∈ X ∧ P x))

Axiom. (exandE_i) We take the following as an axiom:

∀P Q : set → prop, (∃x, P x ∧ Q x) → ∀r : prop, (∀x, P x → Q x → r) → r

Axiom. (exandE_ii) We take the following as an axiom:

∀P Q : (set → set) → prop, (∃x : set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set, P x → Q x → p) → p

Axiom. (exandE_iii) We take the following as an axiom:

∀P Q : (set → set → set) → prop, (∃x : set → set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set → set, P x → Q x → p) → p

Axiom. (exandE_iiii) We take the following as an axiom:

∀P Q : (set → set → set → set) → prop, (∃x : set → set → set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set → set → set, P x → Q x → p) → p

Axiom. (exandE_iio) We take the following as an axiom:

∀P Q : (set → set → prop) → prop, (∃x : set → set → prop, P x ∧ Q x) → ∀p : prop, (∀x : set → set → prop, P x → Q x → p) → p

Axiom. (exandE_iiio) We take the following as an axiom:

∀P Q : (set → set → set → prop) → prop, (∃x : set → set → set → prop, P x ∧ Q x) → ∀p : prop, (∀x : set → set → set → prop, P x → Q x → p) → p

Beginning of Section Descr_ii

Variable P : (set → set) → prop

Primitive. The name Descr_ii is a term of type set → set.

Hypothesis Pex : ∃f : set → set, P f

Hypothesis Puniq : ∀f g : set → set, P f → P g → f = g

Axiom. (Descr_ii_prop) We take the following as an axiom:

P Descr_ii

End of Section Descr_ii

Beginning of Section Descr_iii

Variable P : (set → set → set) → prop

Primitive. The name Descr_iii is a term of type set → set → set.

Hypothesis Pex : ∃f : set → set → set, P f

Hypothesis Puniq : ∀f g : set → set → set, P f → P g → f = g

Axiom. (Descr_iii_prop) We take the following as an axiom:

P Descr_iii

End of Section Descr_iii

Beginning of Section Descr_iio

Variable P : (set → set → prop) → prop

Primitive. The name Descr_iio is a term of type set → set → prop.

Hypothesis Pex : ∃f : set → set → prop, P f

Hypothesis Puniq : ∀f g : set → set → prop, P f → P g → f = g

Axiom. (Descr_iio_prop) We take the following as an axiom:

P Descr_iio

End of Section Descr_iio

Beginning of Section Descr_Vo1

Variable P : Vo 1 → prop

Primitive. The name Descr_Vo1 is a term of type Vo 1.

Hypothesis Pex : ∃f : Vo 1, P f

Hypothesis Puniq : ∀f g : Vo 1, P f → P g → f = g

Axiom. (Descr_Vo1_prop) We take the following as an axiom:

P Descr_Vo1

End of Section Descr_Vo1

Beginning of Section Descr_Vo2

Variable P : Vo 2 → prop

Primitive. The name Descr_Vo2 is a term of type Vo 2.

Hypothesis Pex : ∃f : Vo 2, P f

Hypothesis Puniq : ∀f g : Vo 2, P f → P g → f = g

Axiom. (Descr_Vo2_prop) We take the following as an axiom:

P Descr_Vo2

End of Section Descr_Vo2

Beginning of Section If_ii

Variable p : prop

Variable f g : set → set

Primitive. The name If_ii is a term of type set → set.

Axiom. (If_ii_1) We take the following as an axiom:

p → If_ii = f

Axiom. (If_ii_0) We take the following as an axiom:

¬ p → If_ii = g

End of Section If_ii

Beginning of Section If_iii

Variable p : prop

Variable f g : set → set → set

Primitive. The name If_iii is a term of type set → set → set.

Axiom. (If_iii_1) We take the following as an axiom:

p → If_iii = f

Axiom. (If_iii_0) We take the following as an axiom:

¬ p → If_iii = g

End of Section If_iii

Beginning of Section If_Vo1

Variable p : prop

Variable f g : Vo 1

Primitive. The name If_Vo1 is a term of type Vo 1.

Axiom. (If_Vo1_1) We take the following as an axiom:

p → If_Vo1 = f

Axiom. (If_Vo1_0) We take the following as an axiom:

¬ p → If_Vo1 = g

End of Section If_Vo1

Beginning of Section If_iio

Variable p : prop

Variable f g : set → set → prop

Primitive. The name If_iio is a term of type set → set → prop.

Axiom. (If_iio_1) We take the following as an axiom:

p → If_iio = f

Axiom. (If_iio_0) We take the following as an axiom:

¬ p → If_iio = g

End of Section If_iio

Beginning of Section If_Vo2

Variable p : prop

Variable f g : Vo 2

Primitive. The name If_Vo2 is a term of type Vo 2.

Axiom. (If_Vo2_1) We take the following as an axiom:

p → If_Vo2 = f

Axiom. (If_Vo2_0) We take the following as an axiom:

¬ p → If_Vo2 = g

End of Section If_Vo2

Beginning of Section EpsilonRec_i

Variable F : set → (set → set) → set

Primitive. The name In_rec_i is a term of type set → set.

Hypothesis Fr : ∀X : set, ∀g h : set → set, (∀x ∈ X, g x = h x) → F X g = F X h

Axiom. (In_rec_i_eq) We take the following as an axiom:

∀X : set, In_rec_i X = F X In_rec_i

End of Section EpsilonRec_i

Beginning of Section EpsilonRec_ii

Variable F : set → (set → (set → set)) → (set → set)

Primitive. The name In_rec_ii is a term of type set → (set → set).

Hypothesis Fr : ∀X : set, ∀g h : set → (set → set), (∀x ∈ X, g x = h x) → F X g = F X h

Axiom. (In_rec_ii_eq) We take the following as an axiom:

∀X : set, In_rec_ii X = F X In_rec_ii

End of Section EpsilonRec_ii

Beginning of Section EpsilonRec_iii

Variable F : set → (set → (set → set → set)) → (set → set → set)

Primitive. The name In_rec_iii is a term of type set → (set → set → set).

Hypothesis Fr : ∀X : set, ∀g h : set → (set → set → set), (∀x ∈ X, g x = h x) → F X g = F X h

Axiom. (In_rec_iii_eq) We take the following as an axiom:

∀X : set, In_rec_iii X = F X In_rec_iii

End of Section EpsilonRec_iii

Beginning of Section EpsilonRec_iio

Variable F : set → (set → (set → set → prop)) → (set → set → prop)

Primitive. The name In_rec_iio is a term of type set → (set → set → prop).

Hypothesis Fr : ∀X : set, ∀g h : set → (set → set → prop), (∀x ∈ X, g x = h x) → F X g = F X h

Axiom. (In_rec_iio_eq) We take the following as an axiom:

∀X : set, In_rec_iio X = F X In_rec_iio

End of Section EpsilonRec_iio

Beginning of Section EpsilonRec_Vo1

Variable F : set → (set → Vo 1) → Vo 1

Primitive. The name In_rec_Vo1 is a term of type set → Vo 1.

Hypothesis Fr : ∀X : set, ∀g h : set → Vo 1, (∀x ∈ X, g x = h x) → F X g = F X h

Axiom. (In_rec_Vo1_eq) We take the following as an axiom:

∀X : set, In_rec_Vo1 X = F X In_rec_Vo1

End of Section EpsilonRec_Vo1

Beginning of Section EpsilonRec_Vo2

Variable F : set → (set → Vo 2) → Vo 2

Primitive. The name In_rec_Vo2 is a term of type set → Vo 2.

Hypothesis Fr : ∀X : set, ∀g h : set → Vo 2, (∀x ∈ X, g x = h x) → F X g = F X h

Axiom. (In_rec_Vo2_eq) We take the following as an axiom:

∀X : set, In_rec_Vo2 X = F X In_rec_Vo2

End of Section EpsilonRec_Vo2

Beginning of Section If_Vo3

Variable p : prop

Variable f g : Vo 3

Primitive. The name If_Vo3 is a term of type Vo 3.

Axiom. (If_Vo3_1) We take the following as an axiom:

p → If_Vo3 = f

Axiom. (If_Vo3_0) We take the following as an axiom:

¬ p → If_Vo3 = g

End of Section If_Vo3

Beginning of Section Descr_Vo3

Variable P : Vo 3 → prop

Primitive. The name Descr_Vo3 is a term of type Vo 3.

Hypothesis Pex : ∃f : Vo 3, P f

Hypothesis Puniq : ∀f g : Vo 3, P f → P g → f = g

Axiom. (Descr_Vo3_prop) We take the following as an axiom:

P Descr_Vo3

End of Section Descr_Vo3

Beginning of Section EpsilonRec_Vo3

Variable F : set → (set → Vo 3) → Vo 3

Primitive. The name In_rec_Vo3 is a term of type set → Vo 3.

Hypothesis Fr : ∀X : set, ∀g h : set → Vo 3, (∀x ∈ X, g x = h x) → F X g = F X h

Axiom. (In_rec_Vo3_eq) We take the following as an axiom:

∀X : set, In_rec_Vo3 X = F X In_rec_Vo3

End of Section EpsilonRec_Vo3

Beginning of Section If_Vo4

Variable p : prop

Variable f g : Vo 4

Primitive. The name If_Vo4 is a term of type Vo 4.

Axiom. (If_Vo4_1) We take the following as an axiom:

p → If_Vo4 = f

Axiom. (If_Vo4_0) We take the following as an axiom:

¬ p → If_Vo4 = g

End of Section If_Vo4

Beginning of Section Descr_Vo4

Variable P : Vo 4 → prop

Primitive. The name Descr_Vo4 is a term of type Vo 4.

Hypothesis Pex : ∃f : Vo 4, P f

Hypothesis Puniq : ∀f g : Vo 4, P f → P g → f = g

Axiom. (Descr_Vo4_prop) We take the following as an axiom:

P Descr_Vo4

End of Section Descr_Vo4

Beginning of Section EpsilonRec_Vo4

Variable F : set → (set → Vo 4) → Vo 4

Primitive. The name In_rec_Vo4 is a term of type set → Vo 4.

Hypothesis Fr : ∀X : set, ∀g h : set → Vo 4, (∀x ∈ X, g x = h x) → F X g = F X h

Axiom. (In_rec_Vo4_eq) We take the following as an axiom:

∀X : set, In_rec_Vo4 X = F X In_rec_Vo4

End of Section EpsilonRec_Vo4

Definition. We define bigintersect to be λ(D : (set → prop) → prop)(x : set) ⇒ ∀P : set → prop, D P → P x.

Definition. We define reflexive to be λR ⇒ ∀x : set, R x x of type (set → set → prop) → prop.

Definition. We define irreflexive to be λR ⇒ ∀x : set, ¬ R x x of type (set → set → prop) → prop.

Definition. We define symmetric to be λR ⇒ ∀x y : set, R x y → R y x of type (set → set → prop) → prop.

Definition. We define antisymmetric to be λR ⇒ ∀x y : set, R x y → R y x → x = y of type (set → set → prop) → prop.

Definition. We define transitive to be λR ⇒ ∀x y z : set, R x y → R y z → R x z of type (set → set → prop) → prop.

Definition. We define eqreln to be λR ⇒ reflexive R ∧ symmetric R ∧ transitive R of type (set → set → prop) → prop.

Definition. We define per to be λR ⇒ symmetric R ∧ transitive R of type (set → set → prop) → prop.

Definition. We define linear to be λR ⇒ ∀x y : set, R x y ∨ R y x of type (set → set → prop) → prop.

Definition. We define trichotomous_or to be λR ⇒ ∀x y : set, R x y ∨ x = y ∨ R y x of type (set → set → prop) → prop.

Definition. We define partialorder to be λR ⇒ reflexive R ∧ antisymmetric R ∧ transitive R of type (set → set → prop) → prop.

Definition. We define totalorder to be λR ⇒ partialorder R ∧ linear R of type (set → set → prop) → prop.

Definition. We define strictpartialorder to be λR ⇒ irreflexive R ∧ transitive R of type (set → set → prop) → prop.

Definition. We define stricttotalorder to be λR ⇒ strictpartialorder R ∧ trichotomous_or R of type (set → set → prop) → prop.

Axiom. (per_sym) We take the following as an axiom:

∀R : set → set → prop, per R → symmetric R

Axiom. (per_tra) We take the following as an axiom:

∀R : set → set → prop, per R → transitive R

Axiom. (per_stra1) We take the following as an axiom:

∀R : set → set → prop, per R → ∀x y z : set, R y x → R y z → R x z

Axiom. (per_stra2) We take the following as an axiom:

∀R : set → set → prop, per R → ∀x y z : set, R x y → R z y → R x z

Axiom. (per_stra3) We take the following as an axiom:

∀R : set → set → prop, per R → ∀x y z : set, R y x → R z y → R x z

Axiom. (per_ref1) We take the following as an axiom:

∀R : set → set → prop, per R → ∀x y : set, R x y → R x x

Axiom. (per_ref2) We take the following as an axiom:

∀R : set → set → prop, per R → ∀x y : set, R x y → R y y

Axiom. (partialorder_strictpartialorder) We take the following as an axiom:

∀R : set → set → prop, partialorder R → strictpartialorder (λx y ⇒ R x y ∧ x ≠ y)

Definition. We define reflclos to be λR x y ⇒ R x y ∨ x = y of type (set → set → prop) → (set → set → prop).

Axiom. (reflclos_refl) We take the following as an axiom:

∀R : set → set → prop, reflexive (reflclos R)

Axiom. (reflclos_min) We take the following as an axiom:

∀R S : set → set → prop, R ⊆ S → reflexive S → reflclos R ⊆ S

Axiom. (strictpartialorder_partialorder_reflclos) We take the following as an axiom:

∀R : set → set → prop, strictpartialorder R → partialorder (reflclos R)

Axiom. (stricttotalorder_totalorder_reflclos) We take the following as an axiom:

∀R : set → set → prop, stricttotalorder R → totalorder (reflclos R)

Beginning of Section Zermelo1908

Primitive. The name ZermeloWO is a term of type set → set → prop.

Axiom. (ZermeloWO_Eps) We take the following as an axiom:

∀a : set, (Eps_i (ZermeloWO a)) = a

Axiom. (ZermeloWO_ref) We take the following as an axiom:

reflexive ZermeloWO

Axiom. (ZermeloWO_lin) We take the following as an axiom:

linear ZermeloWO

Axiom. (ZermeloWO_tra) We take the following as an axiom:

transitive ZermeloWO

Axiom. (ZermeloWO_antisym) We take the following as an axiom:

antisymmetric ZermeloWO

Axiom. (ZermeloWO_partialorder) We take the following as an axiom:

partialorder ZermeloWO

Axiom. (ZermeloWO_totalorder) We take the following as an axiom:

totalorder ZermeloWO

Axiom. (ZermeloWO_wo) We take the following as an axiom:

∀p : set → prop, (∃x : set, p x) → ∃x : set, p x ∧ ∀y : set, p y → ZermeloWO x y

Definition. We define ZermeloWOstrict to be λa b : set ⇒ ZermeloWO a b ∧ a ≠ b.

Axiom. (ZermeloWOstrict_trich) We take the following as an axiom:

trichotomous_or ZermeloWOstrict

Axiom. (ZermeloWOstrict_stricttotalorder) We take the following as an axiom:

stricttotalorder ZermeloWOstrict

Axiom. (ZermeloWOstrict_wo) We take the following as an axiom:

∀p : set → prop, (∃x : set, p x) → ∃x : set, p x ∧ ∀y : set, p y ∧ y ≠ x → ZermeloWOstrict x y

Axiom. (Zermelo_WO) We take the following as an axiom:

∃r : set → set → prop, totalorder r ∧ (∀p : set → prop, (∃x : set, p x) → ∃x : set, p x ∧ ∀y : set, p y → r x y)

Axiom. (Zermelo_WO_strict) We take the following as an axiom:

∃r : set → set → prop, stricttotalorder r ∧ (∀p : set → prop, (∃x : set, p x) → ∃x : set, p x ∧ ∀y : set, p y ∧ y ≠ x → r x y)

End of Section Zermelo1908

Axiom. (eq_imp_or) We take the following as an axiom:

(λx y : prop ⇒ (x → y)) = (λx y : prop ⇒ (¬ x ∨ y))

Axiom. (famunion_Empty) We take the following as an axiom:

∀F : set → set, (⋃_{x ∈ 0}F x) = 0

Axiom. (Empty_or_ex) We take the following as an axiom:

∀X : set, X = Empty ∨ ∃x : set, x ∈ X

Axiom. (nIn_0_0) We take the following as an axiom:

0 ∉ 0

Axiom. (nIn_1_0) We take the following as an axiom:

1 ∉ 0

Axiom. (nIn_2_0) We take the following as an axiom:

2 ∉ 0

Axiom. (nIn_1_1) We take the following as an axiom:

1 ∉ 1

Axiom. (nIn_2_2) We take the following as an axiom:

2 ∉ 2

Axiom. (Subq_0_0) We take the following as an axiom:

0 ⊆ 0

Axiom. (Subq_0_1) We take the following as an axiom:

0 ⊆ 1

Axiom. (Subq_0_2) We take the following as an axiom:

0 ⊆ 2

Axiom. (nSubq_1_0) We take the following as an axiom:

1 ⊈ 0

Axiom. (Subq_1_1) We take the following as an axiom:

1 ⊆ 1

Axiom. (Subq_1_2) We take the following as an axiom:

1 ⊆ 2

Axiom. (nSubq_2_0) We take the following as an axiom:

2 ⊈ 0

Axiom. (nSubq_2_1) We take the following as an axiom:

2 ⊈ 1

Axiom. (Subq_2_2) We take the following as an axiom:

2 ⊆ 2

Axiom. (In_0_7) We take the following as an axiom:

0 ∈ 7

Axiom. (In_1_7) We take the following as an axiom:

1 ∈ 7

Axiom. (In_2_7) We take the following as an axiom:

2 ∈ 7

Axiom. (In_3_7) We take the following as an axiom:

3 ∈ 7

Axiom. (In_4_7) We take the following as an axiom:

4 ∈ 7

Axiom. (In_5_7) We take the following as an axiom:

5 ∈ 7

Axiom. (In_6_7) We take the following as an axiom:

6 ∈ 7

Axiom. (In_0_8) We take the following as an axiom:

0 ∈ 8

Axiom. (In_1_8) We take the following as an axiom:

1 ∈ 8

Axiom. (In_2_8) We take the following as an axiom:

2 ∈ 8

Axiom. (In_3_8) We take the following as an axiom:

3 ∈ 8

Axiom. (In_4_8) We take the following as an axiom:

4 ∈ 8

Axiom. (In_5_8) We take the following as an axiom:

5 ∈ 8

Axiom. (In_6_8) We take the following as an axiom:

6 ∈ 8

Axiom. (In_7_8) We take the following as an axiom:

7 ∈ 8

Axiom. (In_0_9) We take the following as an axiom:

0 ∈ 9

Axiom. (In_1_9) We take the following as an axiom:

1 ∈ 9

Axiom. (In_2_9) We take the following as an axiom:

2 ∈ 9

Axiom. (In_3_9) We take the following as an axiom:

3 ∈ 9

Axiom. (In_4_9) We take the following as an axiom:

4 ∈ 9

Axiom. (In_5_9) We take the following as an axiom:

5 ∈ 9

Axiom. (In_6_9) We take the following as an axiom:

6 ∈ 9

Axiom. (In_7_9) We take the following as an axiom:

7 ∈ 9

Axiom. (In_8_9) We take the following as an axiom:

8 ∈ 9

Beginning of Section NatRec

Variable z : set

Variable f : set → set → set

Let F : set → (set → set) → set ≝ λn g ⇒ if ⋃ n ∈ n then f (⋃ n) (g (⋃ n)) else z

Definition. We define nat_primrec to be In_rec_i F of type set → set.

Axiom. (nat_primrec_r) We take the following as an axiom:

∀X : set, ∀g h : set → set, (∀x ∈ X, g x = h x) → F X g = F X h

Axiom. (nat_primrec_0) We take the following as an axiom:

nat_primrec 0 = z

Axiom. (nat_primrec_S) We take the following as an axiom:

∀n : set, nat_p n → nat_primrec (ordsucc n) = f n (nat_primrec n)

End of Section NatRec

Beginning of Section NatArith

Definition. We define add_nat to be λn m : set ⇒ nat_primrec n (λ_ r ⇒ ordsucc r) m of type set → set → set.

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.

Axiom. (add_nat_0R) We take the following as an axiom:

∀n : set, n + 0 = n

Axiom. (add_nat_SR) We take the following as an axiom:

∀n m : set, nat_p m → n + ordsucc m = ordsucc (n + m)

Axiom. (add_nat_p) We take the following as an axiom:

∀n : set, nat_p n → ∀m : set, nat_p m → nat_p (n + m)

Axiom. (add_nat_asso) We take the following as an axiom:

∀n : set, nat_p n → ∀m : set, nat_p m → ∀k : set, nat_p k → (n + m) + k = n + (m + k)

Axiom. (add_nat_0L) We take the following as an axiom:

∀m : set, nat_p m → 0 + m = m

Axiom. (add_nat_SL) We take the following as an axiom:

∀n : set, nat_p n → ∀m : set, nat_p m → ordsucc n + m = ordsucc (n + m)

Axiom. (add_nat_com) We take the following as an axiom:

∀n : set, nat_p n → ∀m : set, nat_p m → n + m = m + n

Definition. We define mul_nat to be λn m : set ⇒ nat_primrec 0 (λ_ r ⇒ n + r) m of type set → set → set.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.

Axiom. (mul_nat_0R) We take the following as an axiom:

∀n : set, n * 0 = 0

Axiom. (mul_nat_SR) We take the following as an axiom:

∀n m : set, nat_p m → n * ordsucc m = n + n * m

Axiom. (mul_nat_p) We take the following as an axiom:

∀n : set, nat_p n → ∀m : set, nat_p m → nat_p (n * m)

Axiom. (mul_nat_0L) We take the following as an axiom:

∀m : set, nat_p m → 0 * m = 0

Axiom. (mul_nat_SL) We take the following as an axiom:

∀n : set, nat_p n → ∀m : set, nat_p m → ordsucc n * m = n * m + m

Axiom. (mul_nat_com) We take the following as an axiom:

∀n : set, nat_p n → ∀m : set, nat_p m → n * m = m * n

Axiom. (mul_add_nat_distrL) We take the following as an axiom:

∀n : set, nat_p n → ∀m : set, nat_p m → ∀k : set, nat_p k → n * (m + k) = n * m + n * k

Axiom. (mul_add_nat_distrR) We take the following as an axiom:

∀n : set, nat_p n → ∀m : set, nat_p m → ∀k : set, nat_p k → (n + m) * k = n * k + m * k

Axiom. (mul_nat_asso) We take the following as an axiom:

∀n : set, nat_p n → ∀m : set, nat_p m → ∀k : set, nat_p k → (n * m) * k = n * (m * k)

Axiom. (add_nat_1_1_2) We take the following as an axiom:

1 + 1 = 2

Definition. We define divides_nat to be λm n ⇒ m ∈ ω ∧ n ∈ ω ∧ ∃k ∈ ω, m * k = n of type set → set → prop.

Definition. We define prime_nat to be λn ⇒ n ∈ ω ∧ 1 ∈ n ∧ ∀k ∈ ω, divides_nat k n → k = 1 ∨ k = n of type set → prop.

Definition. We define coprime_nat to be λa b ⇒ a ∈ ω ∧ b ∈ ω ∧ ∀x ∈ ω ∖ 1, divides_nat x a → divides_nat x b → x = 1 of type set → set → prop.

Definition. We define equiv_nat_mod to be λm k n ⇒ m ∈ ω ∧ k ∈ ω ∧ n ∈ ω ∧ ((∃q ∈ ω, m + q * n = k) ∨ (∃q ∈ ω, k + q * n = m)) of type set → set → set → prop.

Definition. We define exp_nat to be λn m : set ⇒ nat_primrec 1 (λ_ r ⇒ n * r) m of type set → set → set.

Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_nat.

Definition. We define even_nat to be λn ⇒ n ∈ ω ∧ ∃m ∈ ω, n = 2 * m of type set → prop.

Definition. We define odd_nat to be λn ⇒ n ∈ ω ∧ ∀m ∈ ω, n ≠ 2 * m of type set → prop.

Definition. We define nat_factorial to be λn ⇒ nat_primrec 1 (λk r ⇒ ordsucc k * r) n of type set → set.

End of Section NatArith

Axiom. (PigeonHole_nat) We take the following as an axiom:

∀n, nat_p n → ∀f : set → set, (∀i ∈ ordsucc n, f i ∈ n) → ¬ (∀i j ∈ ordsucc n, f i = f j → i = j)

Axiom. (PigeonHole_nat_bij) We take the following as an axiom:

∀n, nat_p n → ∀f : set → set, (∀i ∈ n, f i ∈ n) → (∀i j ∈ n, f i = f j → i = j) → bij n n f

Axiom. (cases_7) We take the following as an axiom:

∀i ∈ 7, ∀p : set → prop, p 0 → p 1 → p 2 → p 3 → p 4 → p 5 → p 6 → p i

Axiom. (cases_8) We take the following as an axiom:

∀i ∈ 8, ∀p : set → prop, p 0 → p 1 → p 2 → p 3 → p 4 → p 5 → p 6 → p 7 → p i

Axiom. (cases_9) We take the following as an axiom:

∀i ∈ 9, ∀p : set → prop, p 0 → p 1 → p 2 → p 3 → p 4 → p 5 → p 6 → p 7 → p 8 → p i

Axiom. (nIn_2_1) We take the following as an axiom:

2 ∉ 1

Axiom. (neq_6_0) We take the following as an axiom:

6 ≠ 0

Axiom. (neq_6_1) We take the following as an axiom:

6 ≠ 1

Axiom. (neq_6_2) We take the following as an axiom:

6 ≠ 2

Axiom. (neq_6_3) We take the following as an axiom:

6 ≠ 3

Axiom. (neq_6_4) We take the following as an axiom:

6 ≠ 4

Axiom. (neq_6_5) We take the following as an axiom:

6 ≠ 5

Axiom. (neq_7_0) We take the following as an axiom:

7 ≠ 0

Axiom. (neq_7_1) We take the following as an axiom:

7 ≠ 1

Axiom. (neq_7_2) We take the following as an axiom:

7 ≠ 2

Axiom. (neq_7_3) We take the following as an axiom:

7 ≠ 3

Axiom. (neq_7_4) We take the following as an axiom:

7 ≠ 4

Axiom. (neq_7_5) We take the following as an axiom:

7 ≠ 5

Axiom. (neq_7_6) We take the following as an axiom:

7 ≠ 6

Axiom. (neq_8_0) We take the following as an axiom:

8 ≠ 0

Axiom. (neq_8_1) We take the following as an axiom:

8 ≠ 1

Axiom. (neq_8_2) We take the following as an axiom:

8 ≠ 2

Axiom. (neq_8_3) We take the following as an axiom:

8 ≠ 3

Axiom. (neq_8_4) We take the following as an axiom:

8 ≠ 4

Axiom. (neq_8_5) We take the following as an axiom:

8 ≠ 5

Axiom. (neq_8_6) We take the following as an axiom:

8 ≠ 6

Axiom. (neq_8_7) We take the following as an axiom:

8 ≠ 7

Axiom. (neq_9_0) We take the following as an axiom:

9 ≠ 0

Axiom. (neq_9_1) We take the following as an axiom:

9 ≠ 1

Axiom. (neq_9_2) We take the following as an axiom:

9 ≠ 2

Axiom. (neq_9_3) We take the following as an axiom:

9 ≠ 3

Axiom. (neq_9_4) We take the following as an axiom:

9 ≠ 4

Axiom. (neq_9_5) We take the following as an axiom:

9 ≠ 5

Axiom. (neq_9_6) We take the following as an axiom:

9 ≠ 6

Axiom. (neq_9_7) We take the following as an axiom:

9 ≠ 7

Axiom. (neq_9_8) We take the following as an axiom:

9 ≠ 8

Axiom. (Subq_1_Sing0) We take the following as an axiom:

1 ⊆ {0}

Axiom. (Subq_Sing0_1) We take the following as an axiom:

{0} ⊆ 1

Axiom. (eq_1_Sing0) We take the following as an axiom:

1 = {0}

Axiom. (Subq_2_UPair01) We take the following as an axiom:

2 ⊆ {0,1}

Axiom. (Subq_UPair01_2) We take the following as an axiom:

{0,1} ⊆ 2

Axiom. (eq_2_UPair01) We take the following as an axiom:

2 = {0,1}

Axiom. (ordinal_ind) We take the following as an axiom:

∀p : set → prop, (∀alpha, ordinal alpha → (∀beta ∈ alpha, p beta) → p alpha) → ∀alpha, ordinal alpha → p alpha

Axiom. (least_ordinal_ex) We take the following as an axiom:

∀p : set → prop, (∃alpha, ordinal alpha ∧ p alpha) → ∃alpha, ordinal alpha ∧ p alpha ∧ ∀beta ∈ alpha, ¬ p beta

Axiom. (ordinal_trichotomy_or_impred) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → ∀p : prop, (alpha ∈ beta → p) → (alpha = beta → p) → (beta ∈ alpha → p) → p

Axiom. (ordinal_trichotomy) We take the following as an axiom:

∀alpha beta : set, ordinal alpha → ordinal beta → exactly1of3 (alpha ∈ beta) (alpha = beta) (beta ∈ alpha)

Definition. We define Inj1 to be In_rec_i (λX f ⇒ {0} ∪ {f x|x ∈ X}) of type set → set.

Axiom. (Inj1_eq) We take the following as an axiom:

∀X : set, Inj1 X = {0} ∪ {Inj1 x|x ∈ X}

Axiom. (Inj1I1) We take the following as an axiom:

∀X : set, 0 ∈ Inj1 X

Axiom. (Inj1I2) We take the following as an axiom:

∀X x : set, x ∈ X → Inj1 x ∈ Inj1 X

Axiom. (Inj1E) We take the following as an axiom:

∀X y : set, y ∈ Inj1 X → y = 0 ∨ ∃x ∈ X, y = Inj1 x

Axiom. (Inj1NE1) We take the following as an axiom:

∀x : set, Inj1 x ≠ 0

Axiom. (Inj1NE2) We take the following as an axiom:

∀x : set, Inj1 x ∉ {0}

Definition. We define Inj0 to be λX ⇒ {Inj1 x|x ∈ X} of type set → set.

Axiom. (Inj0I) We take the following as an axiom:

∀X x : set, x ∈ X → Inj1 x ∈ Inj0 X

Axiom. (Inj0E) We take the following as an axiom:

∀X y : set, y ∈ Inj0 X → ∃x : set, x ∈ X ∧ y = Inj1 x

Definition. We define Unj to be In_rec_i (λX f ⇒ {f x|x ∈ X ∖ {0}}) of type set → set.

Axiom. (Unj_eq) We take the following as an axiom:

∀X : set, Unj X = {Unj x|x ∈ X ∖ {0}}

Axiom. (Unj_Inj1_eq) We take the following as an axiom:

∀X : set, Unj (Inj1 X) = X

Axiom. (Inj1_inj) We take the following as an axiom:

∀X Y : set, Inj1 X = Inj1 Y → X = Y

Axiom. (Unj_Inj0_eq) We take the following as an axiom:

∀X : set, Unj (Inj0 X) = X

Axiom. (Inj0_inj) We take the following as an axiom:

∀X Y : set, Inj0 X = Inj0 Y → X = Y

Axiom. (Inj0_0) We take the following as an axiom:

Inj0 0 = 0

Axiom. (Inj0_Inj1_neq) We take the following as an axiom:

∀X Y : set, Inj0 X ≠ Inj1 Y

Definition. We define setsum to be λX Y ⇒ {Inj0 x|x ∈ X} ∪ {Inj1 y|y ∈ Y} of type set → set → set.

Notation. We use + as an infix operator with priority 450 and which associates to the left corresponding to applying term setsum.

Axiom. (Inj0_setsum) We take the following as an axiom:

∀X Y x : set, x ∈ X → Inj0 x ∈ X + Y

Axiom. (Inj1_setsum) We take the following as an axiom:

∀X Y y : set, y ∈ Y → Inj1 y ∈ X + Y

Axiom. (setsum_Inj_inv) We take the following as an axiom:

∀X Y z : set, z ∈ X + Y → (∃x ∈ X, z = Inj0 x) ∨ (∃y ∈ Y, z = Inj1 y)

Axiom. (Inj0_setsum_0L) We take the following as an axiom:

∀X : set, 0 + X = Inj0 X

Axiom. (Inj1_setsum_1L) We take the following as an axiom:

∀X : set, 1 + X = Inj1 X

Axiom. (nat_setsum1_ordsucc) We take the following as an axiom:

∀n : set, nat_p n → 1 + n = ordsucc n

Axiom. (setsum_0_0) We take the following as an axiom:

0 + 0 = 0

Axiom. (setsum_1_0_1) We take the following as an axiom:

1 + 0 = 1

Axiom. (setsum_1_1_2) We take the following as an axiom:

1 + 1 = 2

Axiom. (setsum_mon) We take the following as an axiom:

∀X Y W Z, X ⊆ W → Y ⊆ Z → X + Y ⊆ W + Z

Definition. We define combine_funcs to be λX Y f g z ⇒ if z = Inj0 (Unj z) then f (Unj z) else g (Unj z) of type set → set → (set → set) → (set → set) → set → set.

Axiom. (combine_funcs_eq1) We take the following as an axiom:

∀X Y, ∀f g : set → set, ∀x, combine_funcs X Y f g (Inj0 x) = f x

Axiom. (combine_funcs_eq2) We take the following as an axiom:

∀X Y, ∀f g : set → set, ∀y, combine_funcs X Y f g (Inj1 y) = g y

Beginning of Section pair_setsum

Let pair ≝ setsum

Axiom. (pair_0_0) We take the following as an axiom:

pair 0 0 = 0

Axiom. (pair_1_0_1) We take the following as an axiom:

pair 1 0 = 1

Axiom. (pair_1_1_2) We take the following as an axiom:

pair 1 1 = 2

Axiom. (nat_pair1_ordsucc) We take the following as an axiom:

∀n : set, nat_p n → pair 1 n = ordsucc n

Definition. We define proj0 to be λZ ⇒ {Unj z|z ∈ Z, ∃x : set, Inj0 x = z} of type set → set.

Definition. We define proj1 to be λZ ⇒ {Unj z|z ∈ Z, ∃y : set, Inj1 y = z} of type set → set.

Axiom. (Inj0_pair_0_eq) We take the following as an axiom:

Inj0 = pair 0

Axiom. (Inj1_pair_1_eq) We take the following as an axiom:

Inj1 = pair 1

Axiom. (pairI0) We take the following as an axiom:

∀X Y x, x ∈ X → pair 0 x ∈ pair X Y

Axiom. (pairI1) We take the following as an axiom:

∀X Y y, y ∈ Y → pair 1 y ∈ pair X Y

Axiom. (pairE) We take the following as an axiom:

∀X Y z, z ∈ pair X Y → (∃x ∈ X, z = pair 0 x) ∨ (∃y ∈ Y, z = pair 1 y)

Axiom. (pairE0) We take the following as an axiom:

∀X Y x, pair 0 x ∈ pair X Y → x ∈ X

Axiom. (pairE1) We take the following as an axiom:

∀X Y y, pair 1 y ∈ pair X Y → y ∈ Y

Axiom. (pairEq) We take the following as an axiom:

∀X Y z, z ∈ pair X Y ↔ (∃x ∈ X, z = pair 0 x) ∨ (∃y ∈ Y, z = pair 1 y)

Axiom. (pairSubq) We take the following as an axiom:

∀X Y W Z, X ⊆ W → Y ⊆ Z → pair X Y ⊆ pair W Z

Axiom. (proj0I) We take the following as an axiom:

∀w u : set, pair 0 u ∈ w → u ∈ proj0 w

Axiom. (proj0E) We take the following as an axiom:

∀w u : set, u ∈ proj0 w → pair 0 u ∈ w

Axiom. (proj1I) We take the following as an axiom:

∀w u : set, pair 1 u ∈ w → u ∈ proj1 w

Axiom. (proj1E) We take the following as an axiom:

∀w u : set, u ∈ proj1 w → pair 1 u ∈ w

Axiom. (proj0_pair_eq) We take the following as an axiom:

∀X Y : set, proj0 (pair X Y) = X

Axiom. (proj1_pair_eq) We take the following as an axiom:

∀X Y : set, proj1 (pair X Y) = Y

Axiom. (pair_inj) We take the following as an axiom:

∀x y w z : set, pair x y = pair w z → x = w ∧ y = z

Axiom. (pair_eta_Subq_proj) We take the following as an axiom:

∀w, pair (proj0 w) (proj1 w) ⊆ w

Definition. We define Sigma to be λX Y ⇒ ⋃_{x ∈ X}{pair x y|y ∈ Y x} of type set → (set → set) → set.

Notation. We use ∑ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Sigma.

Axiom. (pair_Sigma) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀x ∈ X, ∀y ∈ Y x, pair x y ∈ ∑x ∈ X, Y x

Axiom. (Sigma_eta_proj0_proj1) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀z ∈ (∑x ∈ X, Y x), pair (proj0 z) (proj1 z) = z ∧ proj0 z ∈ X ∧ proj1 z ∈ Y (proj0 z)

Axiom. (proj_Sigma_eta) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀z ∈ (∑x ∈ X, Y x), pair (proj0 z) (proj1 z) = z

Axiom. (proj0_Sigma) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → proj0 z ∈ X

Axiom. (proj1_Sigma) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → proj1 z ∈ Y (proj0 z)

Axiom. (pair_Sigma_E0) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀x y : set, pair x y ∈ (∑x ∈ X, Y x) → x ∈ X

Axiom. (pair_Sigma_E1) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀x y : set, pair x y ∈ (∑x ∈ X, Y x) → y ∈ Y x

Axiom. (Sigma_E) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → ∃x ∈ X, ∃y ∈ Y x, z = pair x y

Axiom. (Sigma_Eq) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) ↔ ∃x ∈ X, ∃y ∈ Y x, z = pair x y

Axiom. (Sigma_mon) We take the following as an axiom:

∀X Y : set, X ⊆ Y → ∀Z W : set → set, (∀x ∈ X, Z x ⊆ W x) → (∑x ∈ X, Z x) ⊆ ∑y ∈ Y, W y

Axiom. (Sigma_mon0) We take the following as an axiom:

∀X Y : set, X ⊆ Y → ∀Z : set → set, (∑x ∈ X, Z x) ⊆ ∑y ∈ Y, Z y

Axiom. (Sigma_mon1) We take the following as an axiom:

∀X : set, ∀Z W : set → set, (∀x, x ∈ X → Z x ⊆ W x) → (∑x ∈ X, Z x) ⊆ ∑x ∈ X, W x

Axiom. (Sigma_Power_1) We take the following as an axiom:

∀X : set, X ∈ 𝒫 1 → ∀Y : set → set, (∀x ∈ X, Y x ∈ 𝒫 1) → (∑x ∈ X, Y x) ∈ 𝒫 1

Definition. We define setprod to be λX Y : set ⇒ ∑x ∈ X, Y of type set → set → set.

Notation. We use ⨯ as an infix operator with priority 440 and which associates to the left corresponding to applying term setprod.

Axiom. (pair_setprod) We take the following as an axiom:

∀X Y : set, ∀(x ∈ X)(y ∈ Y), pair x y ∈ X ⨯ Y

Axiom. (proj0_setprod) We take the following as an axiom:

∀X Y : set, ∀z ∈ X ⨯ Y, proj0 z ∈ X

Axiom. (proj1_setprod) We take the following as an axiom:

∀X Y : set, ∀z ∈ X ⨯ Y, proj1 z ∈ Y

Axiom. (pair_setprod_E0) We take the following as an axiom:

∀X Y x y : set, pair x y ∈ X ⨯ Y → x ∈ X

Axiom. (pair_setprod_E1) We take the following as an axiom:

∀X Y x y : set, pair x y ∈ X ⨯ Y → y ∈ Y

Let lam : set → (set → set) → set ≝ Sigma

Definition. We define ap to be λf x ⇒ {proj1 z|z ∈ f, ∃y : set, z = pair x y} of type set → set → set.

Notation. When x is a set, a term x y is notation for ap x y.

Notation. λ x ∈ A ⇒ B is notation for the set Sigma A (λ x : set ⇒ B).

Axiom. (lamI) We take the following as an axiom:

∀X : set, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, pair x y ∈ λx ∈ X ⇒ F x

Axiom. (lamE) We take the following as an axiom:

∀X : set, ∀F : set → set, ∀z : set, z ∈ (λx ∈ X ⇒ F x) → ∃x ∈ X, ∃y ∈ F x, z = pair x y

Axiom. (lamEq) We take the following as an axiom:

∀X : set, ∀F : set → set, ∀z, z ∈ (λx ∈ X ⇒ F x) ↔ ∃x ∈ X, ∃y ∈ F x, z = pair x y

Axiom. (apI) We take the following as an axiom:

∀f x y, pair x y ∈ f → y ∈ f x

Axiom. (apE) We take the following as an axiom:

∀f x y, y ∈ f x → pair x y ∈ f

Axiom. (apEq) We take the following as an axiom:

∀f x y, y ∈ f x ↔ pair x y ∈ f

Axiom. (beta) We take the following as an axiom:

∀X : set, ∀F : set → set, ∀x : set, x ∈ X → (λx ∈ X ⇒ F x) x = F x

Axiom. (beta0) We take the following as an axiom:

∀X : set, ∀F : set → set, ∀x : set, x ∉ X → (λx ∈ X ⇒ F x) x = 0

Axiom. (proj0_ap_0) We take the following as an axiom:

∀u, proj0 u = u 0

Axiom. (proj1_ap_1) We take the following as an axiom:

∀u, proj1 u = u 1

Axiom. (pair_ap_0) We take the following as an axiom:

∀x y : set, (pair x y) 0 = x

Axiom. (pair_ap_1) We take the following as an axiom:

∀x y : set, (pair x y) 1 = y

Axiom. (pair_ap_n2) We take the following as an axiom:

∀x y i : set, i ∉ 2 → (pair x y) i = 0

Axiom. (ap0_Sigma) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → (z 0) ∈ X

Axiom. (ap1_Sigma) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → (z 1) ∈ (Y (z 0))

Definition. We define pair_p to be λu : set ⇒ pair (u 0) (u 1) = u of type set → prop.

Axiom. (pair_p_I) We take the following as an axiom:

∀x y, pair_p (pair x y)

Axiom. (pair_p_I2) We take the following as an axiom:

∀w, (∀u ∈ w, pair_p u ∧ u 0 ∈ 2) → pair_p w

Axiom. (pair_p_In_ap) We take the following as an axiom:

∀w f, pair_p w → w ∈ f → w 1 ∈ ap f (w 0)

Definition. We define tuple_p to be λn u ⇒ ∀z ∈ u, ∃i ∈ n, ∃x : set, z = pair i x of type set → set → prop.

Axiom. (pair_p_tuple2) We take the following as an axiom:

pair_p = tuple_p 2

Axiom. (tuple_p_2_tuple) We take the following as an axiom:

∀x y : set, tuple_p 2 (x,y)

Axiom. (tuple_pair) We take the following as an axiom:

∀x y : set, pair x y = (x,y)

Definition. We define Pi to be λX Y ⇒ {f ∈ 𝒫 (∑x ∈ X, ⋃ (Y x))|∀x ∈ X, f x ∈ Y x} of type set → (set → set) → set.

Notation. We use ∏ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Pi.

Axiom. (PiI) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀f : set, (∀u ∈ f, pair_p u ∧ u 0 ∈ X) → (∀x ∈ X, f x ∈ Y x) → f ∈ ∏x ∈ X, Y x

Axiom. (PiE) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀f : set, f ∈ (∏x ∈ X, Y x) → (∀u ∈ f, pair_p u ∧ u 0 ∈ X) ∧ (∀x ∈ X, f x ∈ Y x)

Axiom. (PiEq) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀f : set, f ∈ Pi X Y ↔ (∀u ∈ f, pair_p u ∧ u 0 ∈ X) ∧ (∀x ∈ X, f x ∈ Y x)

Axiom. (lam_Pi) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀F : set → set, (∀x ∈ X, F x ∈ Y x) → (λx ∈ X ⇒ F x) ∈ (∏x ∈ X, Y x)

Axiom. (ap_Pi) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀f : set, ∀x : set, f ∈ (∏x ∈ X, Y x) → x ∈ X → f x ∈ Y x

Axiom. (Pi_ext_Subq) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀f g ∈ (∏x ∈ X, Y x), (∀x ∈ X, f x ⊆ g x) → f ⊆ g

Axiom. (Pi_ext) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀f g ∈ (∏x ∈ X, Y x), (∀x ∈ X, f x = g x) → f = g

Axiom. (Pi_eta) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀f : set, f ∈ (∏x ∈ X, Y x) → (λx ∈ X ⇒ f x) = f

Definition. We define setexp to be λX Y : set ⇒ ∏y ∈ Y, X of type set → set → set.

Notation. We use :^: as an infix operator with priority 430 and which associates to the left corresponding to applying term setexp.

Axiom. (pair_tuple_fun) We take the following as an axiom:

pair = (λx y ⇒ (x,y))

Axiom. (lamI2) We take the following as an axiom:

∀X, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, (x,y) ∈ λx ∈ X ⇒ F x

Axiom. (lamE2) We take the following as an axiom:

∀X, ∀F : set → set, ∀z : set, z ∈ (λx ∈ X ⇒ F x) → ∃x ∈ X, ∃y ∈ F x, z = (x,y)

Axiom. (tuple_2_inj) We take the following as an axiom:

∀x y w z : set, (x,y) = (w,z) → x = w ∧ y = z

Beginning of Section Tuples

Variable x0 x1 : set

Axiom. (tuple_2_0_eq) We take the following as an axiom:

(x0,x1) 0 = x0

Axiom. (tuple_2_1_eq) We take the following as an axiom:

(x0,x1) 1 = x1

End of Section Tuples

Definition. We define Sep2 to be λX Y R ⇒ {u ∈ ∑x ∈ X, Y x|R (u 0) (u 1)} of type set → (set → set) → (set → set → prop) → set.

Axiom. (Sep2I) We take the following as an axiom:

∀X, ∀Y : set → set, ∀R : set → set → prop, ∀x ∈ X, ∀y ∈ Y x, R x y → (x,y) ∈ Sep2 X Y R

Axiom. (Sep2E) We take the following as an axiom:

∀X, ∀Y : set → set, ∀R : set → set → prop, ∀u ∈ Sep2 X Y R, ∃x ∈ X, ∃y ∈ Y x, u = (x,y) ∧ R x y

Axiom. (Sep2E') We take the following as an axiom:

∀X, ∀Y : set → set, ∀R : set → set → prop, ∀x y, (x,y) ∈ Sep2 X Y R → x ∈ X ∧ y ∈ Y x ∧ R x y

Axiom. (Sep2E'1) We take the following as an axiom:

∀X, ∀Y : set → set, ∀R : set → set → prop, ∀x y, (x,y) ∈ Sep2 X Y R → x ∈ X

Axiom. (Sep2E'2) We take the following as an axiom:

∀X, ∀Y : set → set, ∀R : set → set → prop, ∀x y, (x,y) ∈ Sep2 X Y R → y ∈ Y x

Axiom. (Sep2E'3) We take the following as an axiom:

∀X, ∀Y : set → set, ∀R : set → set → prop, ∀x y, (x,y) ∈ Sep2 X Y R → R x y

Definition. We define set_of_pairs to be λX ⇒ ∀x ∈ X, ∃y z, x = (y,z) of type set → prop.

Axiom. (set_of_pairs_ext) We take the following as an axiom:

∀X Y, set_of_pairs X → set_of_pairs Y → (∀v w, (v,w) ∈ X ↔ (v,w) ∈ Y) → X = Y

Axiom. (Sep2_set_of_pairs) We take the following as an axiom:

∀X, ∀Y : set → set, ∀R : set → set → prop, set_of_pairs (Sep2 X Y R)

Axiom. (Sep2_ext) We take the following as an axiom:

∀X, ∀Y : set → set, ∀R R' : set → set → prop, (∀x ∈ X, ∀y ∈ Y x, R x y ↔ R' x y) → Sep2 X Y R = Sep2 X Y R'

Axiom. (lam_ext_sub) We take the following as an axiom:

∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → (λx ∈ X ⇒ F x) ⊆ (λx ∈ X ⇒ G x)

Axiom. (lam_ext) We take the following as an axiom:

∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → (λx ∈ X ⇒ F x) = (λx ∈ X ⇒ G x)

Axiom. (lam_eta) We take the following as an axiom:

∀X, ∀F : set → set, (λx ∈ X ⇒ (λx ∈ X ⇒ F x) x) = (λx ∈ X ⇒ F x)

Axiom. (tuple_2_eta) We take the following as an axiom:

∀x y, (λi ∈ 2 ⇒ (x,y) i) = (x,y)

Definition. We define lam2 to be λX Y F ⇒ λx ∈ X ⇒ λy ∈ Y x ⇒ F x y of type set → (set → set) → (set → set → set) → set.

Axiom. (beta2) We take the following as an axiom:

∀X, ∀Y : set → set, ∀F : set → set → set, ∀x ∈ X, ∀y ∈ Y x, lam2 X Y F x y = F x y

Axiom. (lam2_ext) We take the following as an axiom:

∀X, ∀Y : set → set, ∀F G : set → set → set, (∀x ∈ X, ∀y ∈ Y x, F x y = G x y) → lam2 X Y F = lam2 X Y G

Definition. We define encode_u to be lam of type set → (set → set) → set.

Definition. We define decode_u to be ap of type set → set → set.

Definition. We define encode_b to be λX F ⇒ lam2 X (λ_ ⇒ X) F of type set → (set → set → set) → set.

Definition. We define decode_b to be λF x y ⇒ F x y of type set → set → set → set.

Definition. We define encode_p to be λX P ⇒ Sep X P of type set → (set → prop) → set.

Definition. We define decode_p to be λP x ⇒ x ∈ P of type set → set → prop.

Definition. We define encode_r to be λX R ⇒ Sep2 X (λ_ ⇒ X) R of type set → (set → set → prop) → set.

Definition. We define decode_r to be λR x y ⇒ (x,y) ∈ R of type set → set → set → prop.

Definition. We define encode_c to be λX C ⇒ Sep (𝒫 X) (λU ⇒ (C (λx ⇒ x ∈ U))) of type set → ((set → prop) → prop) → set.

Definition. We define decode_c to be λC U ⇒ ∃V, (∀x, U x ↔ x ∈ V) ∧ V ∈ C of type set → (set → prop) → prop.

Axiom. (decode_encode_u) We take the following as an axiom:

∀X, ∀F : set → set, ∀x ∈ X, decode_u (encode_u X F) x = F x

Axiom. (encode_u_ext) We take the following as an axiom:

∀X, ∀F F' : set → set, (∀x ∈ X, F x = F' x) → encode_u X F = encode_u X F'

Axiom. (decode_encode_b) We take the following as an axiom:

∀X, ∀F : set → set → set, ∀x y ∈ X, decode_b (encode_b X F) x y = F x y

Axiom. (encode_b_ext) We take the following as an axiom:

∀X, ∀F F' : set → set → set, (∀x y ∈ X, F x y = F' x y) → encode_b X F = encode_b X F'

Axiom. (decode_encode_p) We take the following as an axiom:

∀X, ∀P : set → prop, ∀x ∈ X, (decode_p (encode_p X P) x) = (P x)

Axiom. (encode_p_ext) We take the following as an axiom:

∀X, ∀P P' : set → prop, (∀x ∈ X, P x ↔ P' x) → encode_p X P = encode_p X P'

Axiom. (decode_encode_r) We take the following as an axiom:

∀X, ∀R : set → set → prop, ∀x y ∈ X, (decode_r (encode_r X R) x y) = (R x y)

Axiom. (encode_r_ext) We take the following as an axiom:

∀X, ∀R R' : set → set → prop, (∀x y ∈ X, R x y ↔ R' x y) → encode_r X R = encode_r X R'

Axiom. (decode_encode_c) We take the following as an axiom:

∀X, ∀C : (set → prop) → prop, ∀U : set → prop, (∀x, U x → x ∈ X) → (decode_c (encode_c X C) U) = (C U)

Axiom. (encode_c_ext) We take the following as an axiom:

∀X, ∀C C' : (set → prop) → prop, (∀U : set → prop, (∀x, U x → x ∈ X) → (C U ↔ C' U)) → encode_c X C = encode_c X C'

Axiom. (setprod_mon) We take the following as an axiom:

∀X Y : set, X ⊆ Y → ∀Z W : set, Z ⊆ W → X ⨯ Z ⊆ Y ⨯ W

Axiom. (setprod_mon0) We take the following as an axiom:

∀X Y : set, X ⊆ Y → ∀Z : set, X ⨯ Z ⊆ Y ⨯ Z

Axiom. (setprod_mon1) We take the following as an axiom:

∀X : set, ∀Z W : set, Z ⊆ W → X ⨯ Z ⊆ X ⨯ W

Axiom. (pair_eta_Subq) We take the following as an axiom:

∀w, pair (w 0) (w 1) ⊆ w

Axiom. (Sigma_eta) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀z ∈ (∑x ∈ X, Y x), pair (z 0) (z 1) = z

Axiom. (ReplEq_setprod_ext) We take the following as an axiom:

∀X Y, ∀F G : set → set → set, (∀x ∈ X, ∀y ∈ Y, F x y = G x y) → {F (w 0) (w 1)|w ∈ X ⨯ Y} = {G (w 0) (w 1)|w ∈ X ⨯ Y}

Axiom. (tuple_p_3_tuple) We take the following as an axiom:

∀x y z : set, tuple_p 3 (x,y,z)

Axiom. (tuple_p_4_tuple) We take the following as an axiom:

∀x y z w : set, tuple_p 4 (x,y,z,w)

Axiom. (Pi_Power_1) We take the following as an axiom:

∀X : set, ∀Y : set → set, (∀x ∈ X, Y x ∈ 𝒫 1) → (∏x ∈ X, Y x) ∈ 𝒫 1

Axiom. (Pi_0_dom_mon) We take the following as an axiom:

∀X Y : set, ∀A : set → set, X ⊆ Y → (∀y ∈ Y, y ∉ X → 0 ∈ A y) → (∏x ∈ X, A x) ⊆ ∏y ∈ Y, A y

Axiom. (Pi_cod_mon) We take the following as an axiom:

∀X : set, ∀A B : set → set, (∀x ∈ X, A x ⊆ B x) → (∏x ∈ X, A x) ⊆ ∏x ∈ X, B x

Axiom. (Pi_0_mon) We take the following as an axiom:

∀X Y : set, ∀A B : set → set, (∀x ∈ X, A x ⊆ B x) → X ⊆ Y → (∀y ∈ Y, y ∉ X → 0 ∈ B y) → (∏x ∈ X, A x) ⊆ ∏y ∈ Y, B y

Axiom. (setexp_2_eq) We take the following as an axiom:

∀X : set, X ⨯ X = X²

Axiom. (setexp_0_dom_mon) We take the following as an axiom:

∀A : set, 0 ∈ A → ∀X Y : set, X ⊆ Y → A^X ⊆ A^Y

Axiom. (setexp_0_mon) We take the following as an axiom:

∀X Y A B : set, 0 ∈ B → A ⊆ B → X ⊆ Y → A^X ⊆ B^Y

Axiom. (nat_in_setexp_mon) We take the following as an axiom:

∀A : set, 0 ∈ A → ∀n, nat_p n → ∀m ∈ n, A^m ⊆ Aⁿ

Axiom. (tupleI0) We take the following as an axiom:

∀X Y x, x ∈ X → (0,x) ∈ (X,Y)

Axiom. (tupleI1) We take the following as an axiom:

∀X Y y, y ∈ Y → (1,y) ∈ (X,Y)

Axiom. (tupleE) We take the following as an axiom:

∀X Y z, z ∈ (X,Y) → (∃x ∈ X, z = (0,x)) ∨ (∃y ∈ Y, z = (1,y))

Axiom. (tuple_2_Sigma) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀x ∈ X, ∀y ∈ Y x, (x,y) ∈ ∑x ∈ X, Y x

Axiom. (tuple_2_setprod) We take the following as an axiom:

∀X : set, ∀Y : set, ∀x ∈ X, ∀y ∈ Y, (x,y) ∈ X ⨯ Y

Axiom. (tuple_Sigma_eta) We take the following as an axiom:

∀X : set, ∀Y : set → set, ∀z ∈ (∑x ∈ X, Y x), (z 0,z 1) = z

Axiom. (apI2) We take the following as an axiom:

∀f x y, (x,y) ∈ f → y ∈ f x

Axiom. (apE2) We take the following as an axiom:

∀f x y, y ∈ f x → (x,y) ∈ f

Axiom. (ap_const_0) We take the following as an axiom:

∀x, 0 x = 0

Axiom. (tuple_2_in_A_2) We take the following as an axiom:

∀x y A, x ∈ A → y ∈ A → (x,y) ∈ A²

Axiom. (tuple_2_bij_2) We take the following as an axiom:

∀x y, x ∈ 2 → y ∈ 2 → x ≠ y → bij 2 2 (λi ⇒ (x,y) i)

Axiom. (tuple_3_eta) We take the following as an axiom:

∀x y z, (λi ∈ 3 ⇒ (x,y,z) i) = (x,y,z)

Axiom. (tuple_4_eta) We take the following as an axiom:

∀x y z w, (λi ∈ 4 ⇒ (x,y,z,w) i) = (x,y,z,w)

Beginning of Section Tuples

Variable x0 x1 x2 : set

Axiom. (tuple_3_0_eq) We take the following as an axiom:

(x0,x1,x2) 0 = x0

Axiom. (tuple_3_1_eq) We take the following as an axiom:

(x0,x1,x2) 1 = x1

Axiom. (tuple_3_2_eq) We take the following as an axiom:

(x0,x1,x2) 2 = x2

Variable x3 : set

Axiom. (tuple_4_0_eq) We take the following as an axiom:

(x0,x1,x2,x3) 0 = x0

Axiom. (tuple_4_1_eq) We take the following as an axiom:

(x0,x1,x2,x3) 1 = x1

Axiom. (tuple_4_2_eq) We take the following as an axiom:

(x0,x1,x2,x3) 2 = x2

Axiom. (tuple_4_3_eq) We take the following as an axiom:

(x0,x1,x2,x3) 3 = x3

Variable x4 : set

Axiom. (tuple_5_0_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4) 0 = x0

Axiom. (tuple_5_1_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4) 1 = x1

Axiom. (tuple_5_2_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4) 2 = x2

Axiom. (tuple_5_3_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4) 3 = x3

Axiom. (tuple_5_4_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4) 4 = x4

Variable x5 : set

Axiom. (tuple_6_0_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5) 0 = x0

Axiom. (tuple_6_1_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5) 1 = x1

Axiom. (tuple_6_2_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5) 2 = x2

Axiom. (tuple_6_3_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5) 3 = x3

Axiom. (tuple_6_4_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5) 4 = x4

Axiom. (tuple_6_5_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5) 5 = x5

Variable x6 : set

Axiom. (tuple_7_0_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6) 0 = x0

Axiom. (tuple_7_1_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6) 1 = x1

Axiom. (tuple_7_2_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6) 2 = x2

Axiom. (tuple_7_3_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6) 3 = x3

Axiom. (tuple_7_4_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6) 4 = x4

Axiom. (tuple_7_5_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6) 5 = x5

Axiom. (tuple_7_6_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6) 6 = x6

Variable x7 : set

Axiom. (tuple_8_0_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7) 0 = x0

Axiom. (tuple_8_1_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7) 1 = x1

Axiom. (tuple_8_2_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7) 2 = x2

Axiom. (tuple_8_3_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7) 3 = x3

Axiom. (tuple_8_4_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7) 4 = x4

Axiom. (tuple_8_5_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7) 5 = x5

Axiom. (tuple_8_6_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7) 6 = x6

Axiom. (tuple_8_7_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7) 7 = x7

Variable x8 : set

Axiom. (tuple_9_0_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7,x8) 0 = x0

Axiom. (tuple_9_1_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7,x8) 1 = x1

Axiom. (tuple_9_2_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7,x8) 2 = x2

Axiom. (tuple_9_3_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7,x8) 3 = x3

Axiom. (tuple_9_4_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7,x8) 4 = x4

Axiom. (tuple_9_5_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7,x8) 5 = x5

Axiom. (tuple_9_6_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7,x8) 6 = x6

Axiom. (tuple_9_7_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7,x8) 7 = x7

Axiom. (tuple_9_8_eq) We take the following as an axiom:

(x0,x1,x2,x3,x4,x5,x6,x7,x8) 8 = x8

End of Section Tuples

End of Section pair_setsum

Notation. We use ∑ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Sigma.

Notation. We use ⨯ as an infix operator with priority 440 and which associates to the left corresponding to applying term setprod.

Notation. We use ∏ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Pi.

Notation. We use :^: as an infix operator with priority 430 and which associates to the left corresponding to applying term setexp.

Axiom. (tuple_3_in_A_3) We take the following as an axiom:

∀x y z A, x ∈ A → y ∈ A → z ∈ A → (x,y,z) ∈ A³

Axiom. (tuple_3_bij_3) We take the following as an axiom:

∀x y z, x ∈ 3 → y ∈ 3 → z ∈ 3 → x ≠ y → x ≠ z → y ≠ z → bij 3 3 (λi ⇒ (x,y,z) i)

Axiom. (tuple_4_in_A_4) We take the following as an axiom:

∀x y z w A, x ∈ A → y ∈ A → z ∈ A → w ∈ A → (x,y,z,w) ∈ A⁴

Axiom. (tuple_4_bij_4) We take the following as an axiom:

∀x y z w, x ∈ 4 → y ∈ 4 → z ∈ 4 → w ∈ 4 → x ≠ y → x ≠ z → x ≠ w → y ≠ z → y ≠ w → z ≠ w → bij 4 4 (λi ⇒ (x,y,z,w) i)

Axiom. (iff_refl) We take the following as an axiom:

∀A : prop, A ↔ A

Axiom. (iff_sym) We take the following as an axiom:

∀A B : prop, (A ↔ B) → (B ↔ A)

Axiom. (iff_trans) We take the following as an axiom:

∀A B C : prop, (A ↔ B) → (B ↔ C) → (A ↔ C)

Axiom. (not_or_and_demorgan) We take the following as an axiom:

∀A B : prop, ¬ (A ∨ B) → ¬ A ∧ ¬ B

Axiom. (and_not_or_demorgan) We take the following as an axiom:

∀A B : prop, ¬ A ∧ ¬ B → ¬ (A ∨ B)

Axiom. (not_ex_all_demorgan_i) We take the following as an axiom:

∀P : set → prop, (¬ ∃x, P x) → ∀x, ¬ P x

Axiom. (not_all_ex_demorgan_i) We take the following as an axiom:

∀P : set → prop, ¬ (∀x, P x) → ∃x, ¬ P x

Axiom. (eq_or_nand) We take the following as an axiom:

or = (λx y : prop ⇒ ¬ (¬ x ∧ ¬ y))

Primitive. The name EpsR_i_i_1 is a term of type (set → set → prop) → set.

Primitive. The name EpsR_i_i_2 is a term of type (set → set → prop) → set.

Axiom. (EpsR_i_i_12) We take the following as an axiom:

∀R : set → set → prop, (∃x y, R x y) → R (EpsR_i_i_1 R) (EpsR_i_i_2 R)

Primitive. The name DescrR_i_io_1 is a term of type (set → (set → prop) → prop) → set.

Primitive. The name DescrR_i_io_2 is a term of type (set → (set → prop) → prop) → set → prop.

Axiom. (DescrR_i_io_12) We take the following as an axiom:

∀R : set → (set → prop) → prop, (∃x, (∃y : set → prop, R x y) ∧ (∀y z : set → prop, R x y → R x z → y = z)) → R (DescrR_i_io_1 R) (DescrR_i_io_2 R)

Definition. We define PNoEq_ to be λalpha p q ⇒ ∀beta ∈ alpha, p beta ↔ q beta of type set → (set → prop) → (set → prop) → prop.

Axiom. (PNoEq_ref_) We take the following as an axiom:

∀alpha, ∀p : set → prop, PNoEq_ alpha p p

Axiom. (PNoEq_sym_) We take the following as an axiom:

∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoEq_ alpha q p

Axiom. (PNoEq_tra_) We take the following as an axiom:

∀alpha, ∀p q r : set → prop, PNoEq_ alpha p q → PNoEq_ alpha q r → PNoEq_ alpha p r

Axiom. (PNoEq_antimon_) We take the following as an axiom:

∀p q : set → prop, ∀alpha, ordinal alpha → ∀beta ∈ alpha, PNoEq_ alpha p q → PNoEq_ beta p q

Definition. We define PNoLt_ to be λalpha p q ⇒ ∃beta ∈ alpha, PNoEq_ beta p q ∧ ¬ p beta ∧ q beta of type set → (set → prop) → (set → prop) → prop.

Axiom. (PNoLt_E_) We take the following as an axiom:

∀alpha, ∀p q : set → prop, PNoLt_ alpha p q → ∀R : prop, (∀beta, beta ∈ alpha → PNoEq_ beta p q → ¬ p beta → q beta → R) → R

Axiom. (PNoLt_irref_) We take the following as an axiom:

∀alpha, ∀p : set → prop, ¬ PNoLt_ alpha p p

Axiom. (PNoLt_mon_) We take the following as an axiom:

∀p q : set → prop, ∀alpha, ordinal alpha → ∀beta ∈ alpha, PNoLt_ beta p q → PNoLt_ alpha p q

Axiom. (PNoLt_trichotomy_or_) We take the following as an axiom:

∀p q : set → prop, ∀alpha, ordinal alpha → PNoLt_ alpha p q ∨ PNoEq_ alpha p q ∨ PNoLt_ alpha q p

Axiom. (PNoLt_tra_) We take the following as an axiom:

∀alpha, ordinal alpha → ∀p q r : set → prop, PNoLt_ alpha p q → PNoLt_ alpha q r → PNoLt_ alpha p r

Primitive. The name PNoLt is a term of type set → (set → prop) → set → (set → prop) → prop.

Axiom. (PNoLtI1) We take the following as an axiom:

∀alpha beta, ∀p q : set → prop, PNoLt_ (alpha ∩ beta) p q → PNoLt alpha p beta q

Axiom. (PNoLtI2) We take the following as an axiom:

∀alpha beta, ∀p q : set → prop, alpha ∈ beta → PNoEq_ alpha p q → q alpha → PNoLt alpha p beta q

Axiom. (PNoLtI3) We take the following as an axiom:

∀alpha beta, ∀p q : set → prop, beta ∈ alpha → PNoEq_ beta p q → ¬ p beta → PNoLt alpha p beta q

Axiom. (PNoLtE) We take the following as an axiom:

∀alpha beta, ∀p q : set → prop, PNoLt alpha p beta q → ∀R : prop, (PNoLt_ (alpha ∩ beta) p q → R) → (alpha ∈ beta → PNoEq_ alpha p q → q alpha → R) → (beta ∈ alpha → PNoEq_ beta p q → ¬ p beta → R) → R

Axiom. (PNoLtE2) We take the following as an axiom:

∀alpha, ∀p q : set → prop, PNoLt alpha p alpha q → PNoLt_ alpha p q

Axiom. (PNoLt_irref) We take the following as an axiom:

∀alpha, ∀p : set → prop, ¬ PNoLt alpha p alpha p

Axiom. (PNoLt_trichotomy_or) We take the following as an axiom:

∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNoLt alpha p beta q ∨ alpha = beta ∧ PNoEq_ alpha p q ∨ PNoLt beta q alpha p

Axiom. (PNoLtEq_tra) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → ∀p q r : set → prop, PNoLt alpha p beta q → PNoEq_ beta q r → PNoLt alpha p beta r

Axiom. (PNoEqLt_tra) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → ∀p q r : set → prop, PNoEq_ alpha p q → PNoLt alpha q beta r → PNoLt alpha p beta r

Axiom. (PNoLt_tra) We take the following as an axiom:

∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLt alpha p beta q → PNoLt beta q gamma r → PNoLt alpha p gamma r

Definition. We define PNoLe to be λalpha p beta q ⇒ PNoLt alpha p beta q ∨ alpha = beta ∧ PNoEq_ alpha p q of type set → (set → prop) → set → (set → prop) → prop.

Axiom. (PNoLeI1) We take the following as an axiom:

∀alpha beta, ∀p q : set → prop, PNoLt alpha p beta q → PNoLe alpha p beta q

Axiom. (PNoLeI2) We take the following as an axiom:

∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoLe alpha p alpha q

Axiom. (PNoLe_ref) We take the following as an axiom:

∀alpha, ∀p : set → prop, PNoLe alpha p alpha p

Axiom. (PNoLe_antisym) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → ∀p q : set → prop, PNoLe alpha p beta q → PNoLe beta q alpha p → alpha = beta ∧ PNoEq_ alpha p q

Axiom. (PNoLtLe_tra) We take the following as an axiom:

∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLt alpha p beta q → PNoLe beta q gamma r → PNoLt alpha p gamma r

Axiom. (PNoLeLt_tra) We take the following as an axiom:

∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLe alpha p beta q → PNoLt beta q gamma r → PNoLt alpha p gamma r

Axiom. (PNoEqLe_tra) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → ∀p q r : set → prop, PNoEq_ alpha p q → PNoLe alpha q beta r → PNoLe alpha p beta r

Axiom. (PNoLeEq_tra) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → ∀p q r : set → prop, PNoLe alpha p beta q → PNoEq_ beta q r → PNoLe alpha p beta r

Axiom. (PNoLe_tra) We take the following as an axiom:

∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLe alpha p beta q → PNoLe beta q gamma r → PNoLe alpha p gamma r

Definition. We define PNo_downc to be λL alpha p ⇒ ∃beta, ordinal beta ∧ ∃q : set → prop, L beta q ∧ PNoLe alpha p beta q of type (set → (set → prop) → prop) → set → (set → prop) → prop.

Definition. We define PNo_upc to be λR alpha p ⇒ ∃beta, ordinal beta ∧ ∃q : set → prop, R beta q ∧ PNoLe beta q alpha p of type (set → (set → prop) → prop) → set → (set → prop) → prop.

Axiom. (PNoLe_downc) We take the following as an axiom:

∀L : set → (set → prop) → prop, ∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNo_downc L alpha p → PNoLe beta q alpha p → PNo_downc L beta q

Axiom. (PNo_downc_ref) We take the following as an axiom:

∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, L alpha p → PNo_downc L alpha p

Axiom. (PNo_upc_ref) We take the following as an axiom:

∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, R alpha p → PNo_upc R alpha p

Axiom. (PNoLe_upc) We take the following as an axiom:

∀R : set → (set → prop) → prop, ∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNo_upc R alpha p → PNoLe alpha p beta q → PNo_upc R beta q

Definition. We define PNoLt_pwise to be λL R ⇒ ∀gamma, ordinal gamma → ∀p : set → prop, L gamma p → ∀delta, ordinal delta → ∀q : set → prop, R delta q → PNoLt gamma p delta q of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → prop.

Axiom. (PNoLt_pwise_downc_upc) We take the following as an axiom:

∀L R : set → (set → prop) → prop, PNoLt_pwise L R → PNoLt_pwise (PNo_downc L) (PNo_upc R)

Definition. We define PNo_rel_strict_upperbd to be λL alpha p ⇒ ∀beta ∈ alpha, ∀q : set → prop, PNo_downc L beta q → PNoLt beta q alpha p of type (set → (set → prop) → prop) → set → (set → prop) → prop.

Definition. We define PNo_rel_strict_lowerbd to be λR alpha p ⇒ ∀beta ∈ alpha, ∀q : set → prop, PNo_upc R beta q → PNoLt alpha p beta q of type (set → (set → prop) → prop) → set → (set → prop) → prop.

Definition. We define PNo_rel_strict_imv to be λL R alpha p ⇒ PNo_rel_strict_upperbd L alpha p ∧ PNo_rel_strict_lowerbd R alpha p of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.

Axiom. (PNoEq_rel_strict_upperbd) We take the following as an axiom:

∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_rel_strict_upperbd L alpha p → PNo_rel_strict_upperbd L alpha q

Axiom. (PNo_rel_strict_upperbd_antimon) We take the following as an axiom:

∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, ∀beta ∈ alpha, PNo_rel_strict_upperbd L alpha p → PNo_rel_strict_upperbd L beta p

Axiom. (PNoEq_rel_strict_lowerbd) We take the following as an axiom:

∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_rel_strict_lowerbd R alpha p → PNo_rel_strict_lowerbd R alpha q

Axiom. (PNo_rel_strict_lowerbd_antimon) We take the following as an axiom:

∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, ∀beta ∈ alpha, PNo_rel_strict_lowerbd R alpha p → PNo_rel_strict_lowerbd R beta p

Axiom. (PNoEq_rel_strict_imv) We take the following as an axiom:

∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R alpha q

Axiom. (PNo_rel_strict_imv_antimon) We take the following as an axiom:

∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, ∀beta ∈ alpha, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R beta p

Definition. We define PNo_rel_strict_uniq_imv to be λL R alpha p ⇒ PNo_rel_strict_imv L R alpha p ∧ ∀q : set → prop, PNo_rel_strict_imv L R alpha q → PNoEq_ alpha p q of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.

Definition. We define PNo_rel_strict_split_imv to be λL R alpha p ⇒ PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∧ delta ≠ alpha) ∧ PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∨ delta = alpha) of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.

Axiom. (PNo_extend0_eq) We take the following as an axiom:

∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∧ delta ≠ alpha)

Axiom. (PNo_extend1_eq) We take the following as an axiom:

∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∨ delta = alpha)

Axiom. (PNo_rel_imv_ex) We take the following as an axiom:

∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → (∃p : set → prop, PNo_rel_strict_uniq_imv L R alpha p) ∨ (∃tau ∈ alpha, ∃p : set → prop, PNo_rel_strict_split_imv L R tau p)

Definition. We define PNo_lenbdd to be λalpha L ⇒ ∀beta, ∀p : set → prop, L beta p → beta ∈ alpha of type set → (set → (set → prop) → prop) → prop.

Axiom. (PNo_lenbdd_strict_imv_extend0) We take the following as an axiom:

∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∀p : set → prop, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∧ delta ≠ alpha)

Axiom. (PNo_lenbdd_strict_imv_extend1) We take the following as an axiom:

Axiom. (PNo_lenbdd_strict_imv_split) We take the following as an axiom:

∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∀p : set → prop, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_split_imv L R alpha p

Axiom. (PNo_rel_imv_bdd_ex) We take the following as an axiom:

∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∃beta ∈ ordsucc alpha, ∃p : set → prop, PNo_rel_strict_split_imv L R beta p

Definition. We define PNo_strict_upperbd to be λL alpha p ⇒ ∀beta, ordinal beta → ∀q : set → prop, L beta q → PNoLt beta q alpha p of type (set → (set → prop) → prop) → set → (set → prop) → prop.

Definition. We define PNo_strict_lowerbd to be λR alpha p ⇒ ∀beta, ordinal beta → ∀q : set → prop, R beta q → PNoLt alpha p beta q of type (set → (set → prop) → prop) → set → (set → prop) → prop.

Definition. We define PNo_strict_imv to be λL R alpha p ⇒ PNo_strict_upperbd L alpha p ∧ PNo_strict_lowerbd R alpha p of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.

Axiom. (PNoEq_strict_upperbd) We take the following as an axiom:

∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_strict_upperbd L alpha p → PNo_strict_upperbd L alpha q

Axiom. (PNoEq_strict_lowerbd) We take the following as an axiom:

∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_strict_lowerbd R alpha p → PNo_strict_lowerbd R alpha q

Axiom. (PNoEq_strict_imv) We take the following as an axiom:

∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_strict_imv L R alpha p → PNo_strict_imv L R alpha q

Axiom. (PNo_strict_upperbd_imp_rel_strict_upperbd) We take the following as an axiom:

∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀beta ∈ ordsucc alpha, ∀p : set → prop, PNo_strict_upperbd L alpha p → PNo_rel_strict_upperbd L beta p

Axiom. (PNo_strict_lowerbd_imp_rel_strict_lowerbd) We take the following as an axiom:

∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀beta ∈ ordsucc alpha, ∀p : set → prop, PNo_strict_lowerbd R alpha p → PNo_rel_strict_lowerbd R beta p

Axiom. (PNo_strict_imv_imp_rel_strict_imv) We take the following as an axiom:

∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀beta ∈ ordsucc alpha, ∀p : set → prop, PNo_strict_imv L R alpha p → PNo_rel_strict_imv L R beta p

Axiom. (PNo_rel_split_imv_imp_strict_imv) We take the following as an axiom:

∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, PNo_rel_strict_split_imv L R alpha p → PNo_strict_imv L R alpha p

Axiom. (ordinal_PNo_strict_imv) We take the following as an axiom:

∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, (∀beta ∈ alpha, p beta) → (∀beta, ordinal beta → ∀q : set → prop, L beta q → beta ∈ alpha) → (∀beta ∈ alpha, L beta p) → (∀beta, ordinal beta → ∀q : set → prop, ¬ R beta q) → PNo_strict_imv L R alpha p

Axiom. (PNo_lenbdd_strict_imv_ex) We take the following as an axiom:

∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∃beta ∈ ordsucc alpha, ∃p : set → prop, PNo_strict_imv L R beta p

Definition. We define PNo_least_rep to be λL R beta p ⇒ ordinal beta ∧ PNo_strict_imv L R beta p ∧ ∀gamma ∈ beta, ∀q : set → prop, ¬ PNo_strict_imv L R gamma q of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.

Axiom. (PNo_lenbdd_least_rep_ex) We take the following as an axiom:

∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∃beta, ∃p : set → prop, PNo_least_rep L R beta p

Definition. We define PNo_least_rep2 to be λL R beta p ⇒ PNo_least_rep L R beta p ∧ ∀x, x ∉ beta → ¬ p x of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.

Axiom. (PNo_strict_imv_pred_eq) We take the following as an axiom:

∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → ∀p q : set → prop, PNo_least_rep L R alpha p → PNo_strict_imv L R alpha q → ∀beta ∈ alpha, p beta ↔ q beta

Axiom. (PNo_lenbdd_least_rep2_exuniq2) We take the following as an axiom:

∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∃beta, (∃p : set → prop, PNo_least_rep2 L R beta p) ∧ (∀p q : set → prop, PNo_least_rep2 L R beta p → PNo_least_rep2 L R beta q → p = q)

Primitive. The name PNo_bd is a term of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set.

Primitive. The name PNo_pred is a term of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → prop.

Axiom. (PNo_bd_pred_lem) We take the following as an axiom:

∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → PNo_least_rep2 L R (PNo_bd L R) (PNo_pred L R)

Axiom. (PNo_bd_pred) We take the following as an axiom:

∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → PNo_least_rep L R (PNo_bd L R) (PNo_pred L R)

Axiom. (PNo_bd_ord) We take the following as an axiom:

∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ordinal (PNo_bd L R)

Axiom. (PNo_bd_pred_strict_imv) We take the following as an axiom:

∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → PNo_strict_imv L R (PNo_bd L R) (PNo_pred L R)

Axiom. (PNo_bd_least_imv) We take the following as an axiom:

∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∀gamma ∈ PNo_bd L R, ∀q : set → prop, ¬ PNo_strict_imv L R gamma q

Axiom. (PNo_bd_In) We take the following as an axiom:

∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → PNo_bd L R ∈ ordsucc alpha

Definition. We define PNoCutL to be λalpha p beta q ⇒ beta ∈ alpha ∧ PNoLt beta q alpha p of type set → (set → prop) → set → (set → prop) → prop.

Definition. We define PNoCutR to be λalpha p beta q ⇒ beta ∈ alpha ∧ PNoLt alpha p beta q of type set → (set → prop) → set → (set → prop) → prop.

Axiom. (PNoCutL_lenbdd) We take the following as an axiom:

∀alpha, ∀p : set → prop, PNo_lenbdd alpha (PNoCutL alpha p)

Axiom. (PNoCutR_lenbdd) We take the following as an axiom:

∀alpha, ∀p : set → prop, PNo_lenbdd alpha (PNoCutR alpha p)

Axiom. (PNoCut_pwise) We take the following as an axiom:

∀alpha, ordinal alpha → ∀p : set → prop, PNoLt_pwise (PNoCutL alpha p) (PNoCutR alpha p)

Axiom. (PNoCut_strict_imv) We take the following as an axiom:

∀alpha, ordinal alpha → ∀p : set → prop, PNo_strict_imv (PNoCutL alpha p) (PNoCutR alpha p) alpha p

Axiom. (PNoCut_bd_eq) We take the following as an axiom:

∀alpha, ordinal alpha → ∀p : set → prop, PNo_bd (PNoCutL alpha p) (PNoCutR alpha p) = alpha

Axiom. (PNoCut_pred_eq) We take the following as an axiom:

∀alpha, ordinal alpha → ∀p : set → prop, PNoEq_ alpha p (PNo_pred (PNoCutL alpha p) (PNoCutR alpha p))

Beginning of Section TaggedSets

Let tag : set → set ≝ λalpha ⇒ SetAdjoin alpha {1}

Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.

Axiom. (not_TransSet_Sing1) We take the following as an axiom:

¬ TransSet {1}

Axiom. (not_ordinal_Sing1) We take the following as an axiom:

¬ ordinal {1}

Axiom. (tagged_not_ordinal) We take the following as an axiom:

∀y, ¬ ordinal (y ')

Axiom. (tagged_notin_ordinal) We take the following as an axiom:

∀alpha y, ordinal alpha → (y ') ∉ alpha

Axiom. (tagged_eqE_Subq) We take the following as an axiom:

∀alpha beta, ordinal alpha → alpha ' = beta ' → alpha ⊆ beta

Axiom. (tagged_eqE_eq) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → alpha ' = beta ' → alpha = beta

Axiom. (tagged_ReplE) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → beta ' ∈ {gamma '|gamma ∈ alpha} → beta ∈ alpha

Axiom. (ordinal_notin_tagged_Repl) We take the following as an axiom:

∀alpha Y, ordinal alpha → alpha ∉ {y '|y ∈ Y}

Definition. We define SNoElts_ to be λalpha ⇒ alpha ∪ {beta '|beta ∈ alpha} of type set → set.

Axiom. (SNoElts_mon) We take the following as an axiom:

∀alpha beta, alpha ⊆ beta → SNoElts_ alpha ⊆ SNoElts_ beta

Definition. We define SNo_ to be λalpha x ⇒ x ⊆ SNoElts_ alpha ∧ ∀beta ∈ alpha, exactly1of2 (beta ' ∈ x) (beta ∈ x) of type set → set → prop.

Definition. We define PSNo to be λalpha p ⇒ {beta ∈ alpha|p beta} ∪ {beta '|beta ∈ alpha, ¬ p beta} of type set → (set → prop) → set.

Axiom. (PNoEq_PSNo) We take the following as an axiom:

∀alpha, ordinal alpha → ∀p : set → prop, PNoEq_ alpha (λbeta ⇒ beta ∈ PSNo alpha p) p

Axiom. (SNo_PSNo) We take the following as an axiom:

∀alpha, ordinal alpha → ∀p : set → prop, SNo_ alpha (PSNo alpha p)

Axiom. (SNo_PSNo_eta_) We take the following as an axiom:

∀alpha x, ordinal alpha → SNo_ alpha x → x = PSNo alpha (λbeta ⇒ beta ∈ x)

Primitive. The name SNo is a term of type set → prop.

Axiom. (SNo_SNo) We take the following as an axiom:

∀alpha, ordinal alpha → ∀z, SNo_ alpha z → SNo z

Primitive. The name SNoLev is a term of type set → set.

Axiom. (SNoLev_uniq_Subq) We take the following as an axiom:

∀x alpha beta, ordinal alpha → ordinal beta → SNo_ alpha x → SNo_ beta x → alpha ⊆ beta

Axiom. (SNoLev_uniq) We take the following as an axiom:

∀x alpha beta, ordinal alpha → ordinal beta → SNo_ alpha x → SNo_ beta x → alpha = beta

Axiom. (SNoLev_prop) We take the following as an axiom:

∀x, SNo x → ordinal (SNoLev x) ∧ SNo_ (SNoLev x) x

Axiom. (SNoLev_ordinal) We take the following as an axiom:

∀x, SNo x → ordinal (SNoLev x)

Axiom. (SNoLev_) We take the following as an axiom:

∀x, SNo x → SNo_ (SNoLev x) x

Axiom. (SNo_PSNo_eta) We take the following as an axiom:

∀x, SNo x → x = PSNo (SNoLev x) (λbeta ⇒ beta ∈ x)

Axiom. (SNoLev_PSNo) We take the following as an axiom:

∀alpha, ordinal alpha → ∀p : set → prop, SNoLev (PSNo alpha p) = alpha

Axiom. (SNo_Subq) We take the following as an axiom:

∀x y, SNo x → SNo y → SNoLev x ⊆ SNoLev y → (∀alpha ∈ SNoLev x, alpha ∈ x ↔ alpha ∈ y) → x ⊆ y

Definition. We define SNoEq_ to be λalpha x y ⇒ PNoEq_ alpha (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) of type set → set → set → prop.

Axiom. (SNoEq_I) We take the following as an axiom:

∀alpha x y, (∀beta ∈ alpha, beta ∈ x ↔ beta ∈ y) → SNoEq_ alpha x y

Axiom. (SNoEq_E) We take the following as an axiom:

∀alpha x y, SNoEq_ alpha x y → ∀beta ∈ alpha, beta ∈ x ↔ beta ∈ y

Axiom. (SNoEq_E1) We take the following as an axiom:

∀alpha x y, SNoEq_ alpha x y → ∀beta ∈ alpha, beta ∈ x → beta ∈ y

Axiom. (SNoEq_E2) We take the following as an axiom:

∀alpha x y, SNoEq_ alpha x y → ∀beta ∈ alpha, beta ∈ y → beta ∈ x

Axiom. (SNoEq_antimon_) We take the following as an axiom:

∀alpha, ordinal alpha → ∀beta ∈ alpha, ∀x y, SNoEq_ alpha x y → SNoEq_ beta x y

Axiom. (SNo_eq) We take the following as an axiom:

∀x y, SNo x → SNo y → SNoLev x = SNoLev y → SNoEq_ (SNoLev x) x y → x = y

Let ctag : set → set ≝ λalpha ⇒ SetAdjoin alpha {2}

Notation. We use '' as a postfix operator with priority 100 corresponding to applying term ctag.

Axiom. (ctagged_not_ordinal) We take the following as an axiom:

∀y, ¬ ordinal (y '')

Axiom. (ctagged_notin_ordinal) We take the following as an axiom:

∀alpha y, ordinal alpha → (y '') ∉ alpha

Axiom. (Sing2_notin_SingSing1) We take the following as an axiom:

{2} ∉ {{1}}

Axiom. (ctagged_notin_SNo) We take the following as an axiom:

∀x y, SNo x → (y '') ∉ x

Axiom. (ctagged_eqE_eq) We take the following as an axiom:

∀x y, SNo x → SNo y → ∀u ∈ x, ∀v ∈ y, u '' = v '' → u = v

Definition. We define SNo_pair to be λx y ⇒ x ∪ {u ''|u ∈ y} of type set → set → set.

Axiom. (SNo_pair_prop_1) We take the following as an axiom:

∀x1 y1 x2 y2, SNo x1 → SNo x2 → SNo_pair x1 y1 = SNo_pair x2 y2 → x1 = x2

Axiom. (SNo_pair_prop_2) We take the following as an axiom:

∀x1 y1 x2 y2, SNo x1 → SNo y1 → SNo x2 → SNo y2 → SNo_pair x1 y1 = SNo_pair x2 y2 → y1 = y2

Axiom. (SNo_pair_prop) We take the following as an axiom:

∀x1 y1 x2 y2, SNo x1 → SNo y1 → SNo x2 → SNo y2 → SNo_pair x1 y1 = SNo_pair x2 y2 → x1 = x2 ∧ y1 = y2

Axiom. (SNo_pair_0) We take the following as an axiom:

∀x, SNo_pair x 0 = x

End of Section TaggedSets

Definition. We define SNoLt to be λx y ⇒ PNoLt (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev y) (λbeta ⇒ beta ∈ y) of type set → set → prop.

Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.

Definition. We define SNoLe to be λx y ⇒ PNoLe (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev y) (λbeta ⇒ beta ∈ y) of type set → set → prop.

Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.

Axiom. (SNoLtLe) We take the following as an axiom:

∀x y, x < y → x ≤ y

Axiom. (SNoLeE) We take the following as an axiom:

∀x y, SNo x → SNo y → x ≤ y → x < y ∨ x = y

Axiom. (SNoEq_ref_) We take the following as an axiom:

∀alpha x, SNoEq_ alpha x x

Axiom. (SNoEq_sym_) We take the following as an axiom:

∀alpha x y, SNoEq_ alpha x y → SNoEq_ alpha y x

Axiom. (SNoEq_tra_) We take the following as an axiom:

∀alpha x y z, SNoEq_ alpha x y → SNoEq_ alpha y z → SNoEq_ alpha x z

Axiom. (SNoLtE) We take the following as an axiom:

∀x y, SNo x → SNo y → x < y → ∀p : prop, (∀z, SNo z → SNoLev z ∈ SNoLev x ∩ SNoLev y → SNoEq_ (SNoLev z) z x → SNoEq_ (SNoLev z) z y → x < z → z < y → SNoLev z ∉ x → SNoLev z ∈ y → p) → (SNoLev x ∈ SNoLev y → SNoEq_ (SNoLev x) x y → SNoLev x ∈ y → p) → (SNoLev y ∈ SNoLev x → SNoEq_ (SNoLev y) x y → SNoLev y ∉ x → p) → p

Axiom. (SNoLtI2) We take the following as an axiom:

∀x y, SNoLev x ∈ SNoLev y → SNoEq_ (SNoLev x) x y → SNoLev x ∈ y → x < y

Axiom. (SNoLtI3) We take the following as an axiom:

∀x y, SNoLev y ∈ SNoLev x → SNoEq_ (SNoLev y) x y → SNoLev y ∉ x → x < y

Axiom. (SNoLt_irref) We take the following as an axiom:

∀x, ¬ SNoLt x x

Axiom. (SNoLt_trichotomy_or) We take the following as an axiom:

∀x y, SNo x → SNo y → x < y ∨ x = y ∨ y < x

Axiom. (SNoLt_trichotomy_or_impred) We take the following as an axiom:

∀x y, SNo x → SNo y → ∀p : prop, (x < y → p) → (x = y → p) → (y < x → p) → p

Axiom. (SNoLt_tra) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x < y → y < z → x < z

Axiom. (SNoLe_ref) We take the following as an axiom:

∀x, SNoLe x x

Axiom. (SNoLe_antisym) We take the following as an axiom:

∀x y, SNo x → SNo y → x ≤ y → y ≤ x → x = y

Axiom. (SNoLtLe_tra) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x < y → y ≤ z → x < z

Axiom. (SNoLeLt_tra) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x ≤ y → y < z → x < z

Axiom. (SNoLe_tra) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x ≤ y → y ≤ z → x ≤ z

Axiom. (SNoLtLe_or) We take the following as an axiom:

∀x y, SNo x → SNo y → x < y ∨ y ≤ x

Axiom. (SNoLt_PSNo_PNoLt) We take the following as an axiom:

∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PSNo alpha p < PSNo beta q → PNoLt alpha p beta q

Axiom. (PNoLt_SNoLt_PSNo) We take the following as an axiom:

∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNoLt alpha p beta q → PSNo alpha p < PSNo beta q

Definition. We define SNoCut to be λL R ⇒ PSNo (PNo_bd (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ L) (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ R)) (PNo_pred (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ L) (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ R)) of type set → set → set.

Definition. We define SNoCutP to be λL R ⇒ (∀x ∈ L, SNo x) ∧ (∀y ∈ R, SNo y) ∧ (∀x ∈ L, ∀y ∈ R, x < y) of type set → set → prop.

Axiom. (SNoCutP_SNoCut) We take the following as an axiom:

∀L R, SNoCutP L R → SNo (SNoCut L R) ∧ SNoLev (SNoCut L R) ∈ ordsucc ((⋃_{x ∈ L}ordsucc (SNoLev x)) ∪ (⋃_{y ∈ R}ordsucc (SNoLev y))) ∧ (∀x ∈ L, x < SNoCut L R) ∧ (∀y ∈ R, SNoCut L R < y) ∧ (∀z, SNo z → (∀x ∈ L, x < z) → (∀y ∈ R, z < y) → SNoLev (SNoCut L R) ⊆ SNoLev z ∧ SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z)

Axiom. (SNoCutP_SNoCut_impred) We take the following as an axiom:

∀L R, SNoCutP L R → ∀p : prop, (SNo (SNoCut L R) → SNoLev (SNoCut L R) ∈ ordsucc ((⋃_{x ∈ L}ordsucc (SNoLev x)) ∪ (⋃_{y ∈ R}ordsucc (SNoLev y))) → (∀x ∈ L, x < SNoCut L R) → (∀y ∈ R, SNoCut L R < y) → (∀z, SNo z → (∀x ∈ L, x < z) → (∀y ∈ R, z < y) → SNoLev (SNoCut L R) ⊆ SNoLev z ∧ SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z) → p) → p

Axiom. (SNoCutP_L_0) We take the following as an axiom:

∀L, (∀x ∈ L, SNo x) → SNoCutP L 0

Axiom. (SNoCutP_0_R) We take the following as an axiom:

∀R, (∀x ∈ R, SNo x) → SNoCutP 0 R

Axiom. (SNoCutP_0_0) We take the following as an axiom:

SNoCutP 0 0

Axiom. (SNoCut_0_0) We take the following as an axiom:

SNoCut 0 0 = 0

Axiom. (ordinal_SNoLt_In) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → alpha < beta → alpha ∈ beta

Axiom. (ordinal_SNoLe_Subq) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → alpha ≤ beta → alpha ⊆ beta

Definition. We define SNoS_ to be λalpha ⇒ {x ∈ 𝒫 (SNoElts_ alpha)|∃beta ∈ alpha, SNo_ beta x} of type set → set.

Axiom. (SNoS_E) We take the following as an axiom:

∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, ∃beta ∈ alpha, SNo_ beta x

Beginning of Section TaggedSets2

Let tag : set → set ≝ λalpha ⇒ SetAdjoin alpha {1}

Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.

Axiom. (SNoS_I) We take the following as an axiom:

∀alpha, ordinal alpha → ∀x, ∀beta ∈ alpha, SNo_ beta x → x ∈ SNoS_ alpha

Axiom. (SNoS_I2) We take the following as an axiom:

∀x y, SNo x → SNo y → SNoLev x ∈ SNoLev y → x ∈ SNoS_ (SNoLev y)

Axiom. (SNoS_Subq) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta → SNoS_ alpha ⊆ SNoS_ beta

Axiom. (SNoLev_uniq2) We take the following as an axiom:

∀alpha, ordinal alpha → ∀x, SNo_ alpha x → SNoLev x = alpha

Axiom. (SNoS_E2) We take the following as an axiom:

∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, ∀p : prop, (SNoLev x ∈ alpha → ordinal (SNoLev x) → SNo x → SNo_ (SNoLev x) x → p) → p

Axiom. (SNoS_In_neq) We take the following as an axiom:

∀w, SNo w → ∀x ∈ SNoS_ (SNoLev w), x ≠ w

Axiom. (SNoS_SNoLev) We take the following as an axiom:

∀z, SNo z → z ∈ SNoS_ (ordsucc (SNoLev z))

Definition. We define SNoL to be λz ⇒ {x ∈ SNoS_ (SNoLev z)|x < z} of type set → set.

Definition. We define SNoR to be λz ⇒ {y ∈ SNoS_ (SNoLev z)|z < y} of type set → set.

Axiom. (SNoCutP_SNoL_SNoR) We take the following as an axiom:

∀z, SNo z → SNoCutP (SNoL z) (SNoR z)

Axiom. (SNoL_E) We take the following as an axiom:

∀x, SNo x → ∀w ∈ SNoL x, ∀p : prop, (SNo w → SNoLev w ∈ SNoLev x → w < x → p) → p

Axiom. (SNoR_E) We take the following as an axiom:

∀x, SNo x → ∀z ∈ SNoR x, ∀p : prop, (SNo z → SNoLev z ∈ SNoLev x → x < z → p) → p

Axiom. (SNoL_SNoS) We take the following as an axiom:

∀x, SNo x → ∀w ∈ SNoL x, w ∈ SNoS_ (SNoLev x)

Axiom. (SNoR_SNoS) We take the following as an axiom:

∀x, SNo x → ∀z ∈ SNoR x, z ∈ SNoS_ (SNoLev x)

Axiom. (SNoL_SNoS_) We take the following as an axiom:

∀z, SNoL z ⊆ SNoS_ (SNoLev z)

Axiom. (SNoR_SNoS_) We take the following as an axiom:

∀z, SNoR z ⊆ SNoS_ (SNoLev z)

Axiom. (SNoL_I) We take the following as an axiom:

∀z, SNo z → ∀x, SNo x → SNoLev x ∈ SNoLev z → x < z → x ∈ SNoL z

Axiom. (SNoR_I) We take the following as an axiom:

∀z, SNo z → ∀y, SNo y → SNoLev y ∈ SNoLev z → z < y → y ∈ SNoR z

Axiom. (SNo_eta) We take the following as an axiom:

∀z, SNo z → z = SNoCut (SNoL z) (SNoR z)

Axiom. (SNoCutP_SNo_SNoCut) We take the following as an axiom:

∀L R, SNoCutP L R → SNo (SNoCut L R)

Axiom. (SNoCutP_SNoCut_L) We take the following as an axiom:

∀L R, SNoCutP L R → ∀x ∈ L, x < SNoCut L R

Axiom. (SNoCutP_SNoCut_R) We take the following as an axiom:

∀L R, SNoCutP L R → ∀y ∈ R, SNoCut L R < y

Axiom. (SNoCutP_SNoCut_fst) We take the following as an axiom:

∀L R, SNoCutP L R → ∀z, SNo z → (∀x ∈ L, x < z) → (∀y ∈ R, z < y) → SNoLev (SNoCut L R) ⊆ SNoLev z ∧ SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z

Axiom. (SNoCut_Le) We take the following as an axiom:

∀L1 R1 L2 R2, SNoCutP L1 R1 → SNoCutP L2 R2 → (∀w ∈ L1, w < SNoCut L2 R2) → (∀z ∈ R2, SNoCut L1 R1 < z) → SNoCut L1 R1 ≤ SNoCut L2 R2

Axiom. (SNoCut_ext) We take the following as an axiom:

∀L1 R1 L2 R2, SNoCutP L1 R1 → SNoCutP L2 R2 → (∀w ∈ L1, w < SNoCut L2 R2) → (∀z ∈ R1, SNoCut L2 R2 < z) → (∀w ∈ L2, w < SNoCut L1 R1) → (∀z ∈ R2, SNoCut L1 R1 < z) → SNoCut L1 R1 = SNoCut L2 R2

Axiom. (SNoLt_SNoL_or_SNoR_impred) We take the following as an axiom:

∀x y, SNo x → SNo y → x < y → ∀p : prop, (∀z ∈ SNoL y, z ∈ SNoR x → p) → (x ∈ SNoL y → p) → (y ∈ SNoR x → p) → p

Axiom. (SNoL_or_SNoR_impred) We take the following as an axiom:

∀x y, SNo x → SNo y → ∀p : prop, (x = y → p) → (∀z ∈ SNoL y, z ∈ SNoR x → p) → (x ∈ SNoL y → p) → (y ∈ SNoR x → p) → (∀z ∈ SNoR y, z ∈ SNoL x → p) → (x ∈ SNoR y → p) → (y ∈ SNoL x → p) → p

Axiom. (ordinal_SNo_) We take the following as an axiom:

∀alpha, ordinal alpha → SNo_ alpha alpha

Axiom. (ordinal_SNoL) We take the following as an axiom:

∀alpha, ordinal alpha → SNoL alpha = SNoS_ alpha

Axiom. (ordinal_SNoR) We take the following as an axiom:

∀alpha, ordinal alpha → SNoR alpha = Empty

Axiom. (ordinal_SNoCutP) We take the following as an axiom:

∀alpha, ordinal alpha → SNoCutP (SNoS_ alpha) Empty

Axiom. (ordinal_SNoCut_eta) We take the following as an axiom:

∀alpha, ordinal alpha → alpha = SNoCut (SNoS_ alpha) Empty

Axiom. (SNo_0) We take the following as an axiom:

SNo 0

Axiom. (SNoLev_0) We take the following as an axiom:

SNoLev 0 = 0

Axiom. (SNoL_0) We take the following as an axiom:

SNoL 0 = 0

Axiom. (SNoR_0) We take the following as an axiom:

SNoR 0 = 0

Axiom. (SNoL_1) We take the following as an axiom:

SNoL 1 = 1

Axiom. (SNoR_1) We take the following as an axiom:

SNoR 1 = 0

Axiom. (SNo_max_SNoLev) We take the following as an axiom:

∀x, SNo x → (∀y ∈ SNoS_ (SNoLev x), y < x) → SNoLev x = x

Axiom. (SNo_max_ordinal) We take the following as an axiom:

∀x, SNo x → (∀y ∈ SNoS_ (SNoLev x), y < x) → ordinal x

Definition. We define SNo_extend0 to be λx ⇒ PSNo (ordsucc (SNoLev x)) (λdelta ⇒ delta ∈ x ∧ delta ≠ SNoLev x) of type set → set.

Definition. We define SNo_extend1 to be λx ⇒ PSNo (ordsucc (SNoLev x)) (λdelta ⇒ delta ∈ x ∨ delta = SNoLev x) of type set → set.

Axiom. (SNo_extend0_SNo_) We take the following as an axiom:

∀x, SNo x → SNo_ (ordsucc (SNoLev x)) (SNo_extend0 x)

Axiom. (SNo_extend1_SNo_) We take the following as an axiom:

∀x, SNo x → SNo_ (ordsucc (SNoLev x)) (SNo_extend1 x)

Axiom. (SNo_extend0_SNo) We take the following as an axiom:

∀x, SNo x → SNo (SNo_extend0 x)

Axiom. (SNo_extend1_SNo) We take the following as an axiom:

∀x, SNo x → SNo (SNo_extend1 x)

Axiom. (SNo_extend0_SNoLev) We take the following as an axiom:

∀x, SNo x → SNoLev (SNo_extend0 x) = ordsucc (SNoLev x)

Axiom. (SNo_extend1_SNoLev) We take the following as an axiom:

∀x, SNo x → SNoLev (SNo_extend1 x) = ordsucc (SNoLev x)

Axiom. (SNo_extend0_nIn) We take the following as an axiom:

∀x, SNo x → SNoLev x ∉ SNo_extend0 x

Axiom. (SNo_extend1_In) We take the following as an axiom:

∀x, SNo x → SNoLev x ∈ SNo_extend1 x

Axiom. (SNo_extend0_SNoEq) We take the following as an axiom:

∀x, SNo x → SNoEq_ (SNoLev x) (SNo_extend0 x) x

Axiom. (SNo_extend1_SNoEq) We take the following as an axiom:

∀x, SNo x → SNoEq_ (SNoLev x) (SNo_extend1 x) x

Axiom. (SNo_extend0_Lt) We take the following as an axiom:

∀x, SNo x → SNo_extend0 x < x

Axiom. (SNo_extend1_Gt) We take the following as an axiom:

∀x, SNo x → x < SNo_extend1 x

Definition. We define eps_ to be λn ⇒ {0} ∪ {(ordsucc m) '|m ∈ n} of type set → set.

Axiom. (eps_ordinal_In_eq_0) We take the following as an axiom:

∀n alpha, ordinal alpha → alpha ∈ eps_ n → alpha = 0

Axiom. (eps_0_1) We take the following as an axiom:

eps_ 0 = 1

Axiom. (SNo__eps_) We take the following as an axiom:

∀n ∈ ω, SNo_ (ordsucc n) (eps_ n)

Axiom. (SNo_eps_) We take the following as an axiom:

∀n ∈ ω, SNo (eps_ n)

Axiom. (SNoLev_eps_) We take the following as an axiom:

∀n ∈ ω, SNoLev (eps_ n) = ordsucc n

Axiom. (SNo_eps_SNoS_omega) We take the following as an axiom:

∀n ∈ ω, eps_ n ∈ SNoS_ ω

Axiom. (SNo_eps_decr) We take the following as an axiom:

∀n ∈ ω, ∀m ∈ n, eps_ n < eps_ m

Axiom. (SNo_eps_pos) We take the following as an axiom:

∀n ∈ ω, 0 < eps_ n

Axiom. (SNo_pos_eps_Lt) We take the following as an axiom:

∀n, nat_p n → ∀x ∈ SNoS_ n, 0 < x → eps_ n < x

End of Section TaggedSets2

Axiom. (ordinal_SNo) We take the following as an axiom:

∀alpha, ordinal alpha → SNo alpha

Axiom. (ordinal_SNoLev) We take the following as an axiom:

∀alpha, ordinal alpha → SNoLev alpha = alpha

Axiom. (ordinal_SNoLev_max) We take the following as an axiom:

∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ alpha → z < alpha

Axiom. (ordinal_In_SNoLt) We take the following as an axiom:

∀alpha, ordinal alpha → ∀beta ∈ alpha, beta < alpha

Axiom. (ordinal_SNoLev_max_2) We take the following as an axiom:

∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ ordsucc alpha → z ≤ alpha

Axiom. (ordinal_Subq_SNoLe) We take the following as an axiom:

∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta → alpha ≤ beta

Axiom. (SNo_etaE) We take the following as an axiom:

∀z, SNo z → ∀p : prop, (∀L R, SNoCutP L R → (∀x ∈ L, SNoLev x ∈ SNoLev z) → (∀y ∈ R, SNoLev y ∈ SNoLev z) → z = SNoCut L R → p) → p

Axiom. (SNo_ind) We take the following as an axiom:

∀P : set → prop, (∀L R, SNoCutP L R → (∀x ∈ L, P x) → (∀y ∈ R, P y) → P (SNoCut L R)) → ∀z, SNo z → P z

Beginning of Section SurrealRecI

Variable F : set → (set → set) → set

Primitive. The name SNo_rec_i is a term of type set → set.

Hypothesis Fr : ∀z, SNo z → ∀g h : set → set, (∀w ∈ SNoS_ (SNoLev z), g w = h w) → F z g = F z h

Axiom. (SNo_rec_i_eq) We take the following as an axiom:

∀z, SNo z → SNo_rec_i z = F z SNo_rec_i

End of Section SurrealRecI

Beginning of Section SurrealRecII

Variable F : set → (set → (set → set)) → (set → set)

Primitive. The name SNo_rec_ii is a term of type set → (set → set).

Hypothesis Fr : ∀z, SNo z → ∀g h : set → (set → set), (∀w ∈ SNoS_ (SNoLev z), g w = h w) → F z g = F z h

Axiom. (SNo_rec_ii_eq) We take the following as an axiom:

∀z, SNo z → SNo_rec_ii z = F z SNo_rec_ii

End of Section SurrealRecII

Beginning of Section SurrealRec2

Variable F : set → set → (set → set → set) → set

Primitive. The name SNo_rec2 is a term of type set → set → set.

Hypothesis Fr : ∀w, SNo w → ∀z, SNo z → ∀g h : set → set → set, (∀x ∈ SNoS_ (SNoLev w), ∀y, SNo y → g x y = h x y) → (∀y ∈ SNoS_ (SNoLev z), g w y = h w y) → F w z g = F w z h

Axiom. (SNo_rec2_eq) We take the following as an axiom:

∀w, SNo w → ∀z, SNo z → SNo_rec2 w z = F w z SNo_rec2

End of Section SurrealRec2

Axiom. (SNo_ordinal_ind) We take the following as an axiom:

∀P : set → prop, (∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, P x) → (∀x, SNo x → P x)

Axiom. (SNo_ordinal_ind2) We take the following as an axiom:

∀P : set → set → prop, (∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀x ∈ SNoS_ alpha, ∀y ∈ SNoS_ beta, P x y) → (∀x y, SNo x → SNo y → P x y)

Axiom. (SNo_ordinal_ind3) We take the following as an axiom:

∀P : set → set → set → prop, (∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀gamma, ordinal gamma → ∀x ∈ SNoS_ alpha, ∀y ∈ SNoS_ beta, ∀z ∈ SNoS_ gamma, P x y z) → (∀x y z, SNo x → SNo y → SNo z → P x y z)

Axiom. (SNoLev_ind) We take the following as an axiom:

∀P : set → prop, (∀x, SNo x → (∀w ∈ SNoS_ (SNoLev x), P w) → P x) → (∀x, SNo x → P x)

Axiom. (SNoLev_ind2) We take the following as an axiom:

∀P : set → set → prop, (∀x y, SNo x → SNo y → (∀w ∈ SNoS_ (SNoLev x), P w y) → (∀z ∈ SNoS_ (SNoLev y), P x z) → (∀w ∈ SNoS_ (SNoLev x), ∀z ∈ SNoS_ (SNoLev y), P w z) → P x y) → ∀x y, SNo x → SNo y → P x y

Axiom. (SNoLev_ind3) We take the following as an axiom:

∀P : set → set → set → prop, (∀x y z, SNo x → SNo y → SNo z → (∀u ∈ SNoS_ (SNoLev x), P u y z) → (∀v ∈ SNoS_ (SNoLev y), P x v z) → (∀w ∈ SNoS_ (SNoLev z), P x y w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), P u v z) → (∀u ∈ SNoS_ (SNoLev x), ∀w ∈ SNoS_ (SNoLev z), P u y w) → (∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), P x v w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), P u v w) → P x y z) → ∀x y z, SNo x → SNo y → SNo z → P x y z

Axiom. (SNo_1) We take the following as an axiom:

SNo 1

Axiom. (SNo_2) We take the following as an axiom:

SNo 2

Axiom. (SNo_omega) We take the following as an axiom:

SNo ω

Axiom. (SNoLt_0_1) We take the following as an axiom:

0 < 1

Axiom. (SNoLt_0_2) We take the following as an axiom:

0 < 2

Axiom. (SNoLt_1_2) We take the following as an axiom:

1 < 2

Axiom. (SNoLev_0_eq_0) We take the following as an axiom:

∀x, SNo x → SNoLev x = 0 → x = 0

Axiom. (restr_SNo_) We take the following as an axiom:

∀x, SNo x → ∀alpha ∈ SNoLev x, SNo_ alpha (x ∩ SNoElts_ alpha)

Axiom. (restr_SNo) We take the following as an axiom:

∀x, SNo x → ∀alpha ∈ SNoLev x, SNo (x ∩ SNoElts_ alpha)

Axiom. (restr_SNoLev) We take the following as an axiom:

∀x, SNo x → ∀alpha ∈ SNoLev x, SNoLev (x ∩ SNoElts_ alpha) = alpha

Axiom. (restr_SNoEq) We take the following as an axiom:

∀x, SNo x → ∀alpha ∈ SNoLev x, SNoEq_ alpha (x ∩ SNoElts_ alpha) x

Axiom. (restr_SNo_SNoCut) We take the following as an axiom:

∀x, SNo x → ∀alpha ∈ SNoLev x, ∀p : prop, (SNoCutP {w ∈ SNoL x|SNoLev w ∈ alpha} {z ∈ SNoR x|SNoLev z ∈ alpha} → x ∩ SNoElts_ alpha = SNoCut {w ∈ SNoL x|SNoLev w ∈ alpha} {z ∈ SNoR x|SNoLev z ∈ alpha} → p) → p

Primitive. The name pack_e is a term of type set → set → set.

Axiom. (pack_e_0_eq) We take the following as an axiom:

∀S X, ∀c : set, S = pack_e X c → X = S 0

Axiom. (pack_e_0_eq2) We take the following as an axiom:

∀X, ∀c : set, X = pack_e X c 0

Axiom. (pack_e_1_eq) We take the following as an axiom:

∀S X, ∀c : set, S = pack_e X c → c = S 1

Axiom. (pack_e_1_eq2) We take the following as an axiom:

∀X, ∀c : set, c = pack_e X c 1

Axiom. (pack_e_inj) We take the following as an axiom:

∀X X', ∀c c', pack_e X c = pack_e X' c' → X = X' ∧ c = c'

Definition. We define struct_e to be λS ⇒ ∀q : set → prop, (∀X : set, ∀c : set, c ∈ X → q (pack_e X c)) → q S of type set → prop.

Axiom. (pack_struct_e_I) We take the following as an axiom:

∀X, ∀c : set, c ∈ X → struct_e (pack_e X c)

Axiom. (pack_struct_e_E1) We take the following as an axiom:

∀X, ∀c : set, struct_e (pack_e X c) → c ∈ X

Axiom. (struct_e_eta) We take the following as an axiom:

∀S, struct_e S → S = pack_e (S 0) (S 1)

Primitive. The name unpack_e_i is a term of type set → (set → set → set) → set.

Axiom. (unpack_e_i_eq) We take the following as an axiom:

∀Phi : set → set → set, ∀X, ∀c : set, unpack_e_i (pack_e X c) Phi = Phi X c

Primitive. The name unpack_e_o is a term of type set → (set → set → prop) → prop.

Axiom. (unpack_e_o_eq) We take the following as an axiom:

∀Phi : set → set → prop, ∀X, ∀c : set, unpack_e_o (pack_e X c) Phi = Phi X c

Primitive. The name pack_u is a term of type set → (set → set) → set.

Axiom. (pack_u_0_eq) We take the following as an axiom:

∀S X, ∀F : set → set, S = pack_u X F → X = S 0

Axiom. (pack_u_0_eq2) We take the following as an axiom:

∀X, ∀F : set → set, X = pack_u X F 0

Axiom. (pack_u_1_eq) We take the following as an axiom:

∀S X, ∀F : set → set, S = pack_u X F → ∀x ∈ X, F x = decode_u (S 1) x

Axiom. (pack_u_1_eq2) We take the following as an axiom:

∀X, ∀F : set → set, ∀x ∈ X, F x = decode_u (pack_u X F 1) x

Axiom. (pack_u_inj) We take the following as an axiom:

∀X X', ∀F F' : set → set, pack_u X F = pack_u X' F' → X = X' ∧ ∀x ∈ X, F x = F' x

Axiom. (pack_u_ext) We take the following as an axiom:

∀X, ∀F F' : set → set, (∀x ∈ X, F x = F' x) → pack_u X F = pack_u X F'

Definition. We define struct_u to be λS ⇒ ∀q : set → prop, (∀X, ∀F : set → set, (∀x ∈ X, F x ∈ X) → q (pack_u X F)) → q S of type set → prop.

Axiom. (pack_struct_u_I) We take the following as an axiom:

∀X, ∀F : set → set, (∀x ∈ X, F x ∈ X) → struct_u (pack_u X F)

Axiom. (pack_struct_u_E1) We take the following as an axiom:

∀X, ∀F : set → set, struct_u (pack_u X F) → ∀x ∈ X, F x ∈ X

Axiom. (struct_u_eta) We take the following as an axiom:

∀S, struct_u S → S = pack_u (S 0) (decode_u (S 1))

Primitive. The name unpack_u_i is a term of type set → (set → (set → set) → set) → set.

Axiom. (unpack_u_i_eq) We take the following as an axiom:

∀Phi : set → (set → set) → set, ∀X, ∀F : set → set, (∀F' : set → set, (∀x ∈ X, F x = F' x) → Phi X F' = Phi X F) → unpack_u_i (pack_u X F) Phi = Phi X F

Primitive. The name unpack_u_o is a term of type set → (set → (set → set) → prop) → prop.

Axiom. (unpack_u_o_eq) We take the following as an axiom:

∀Phi : set → (set → set) → prop, ∀X, ∀F : set → set, (∀F' : set → set, (∀x ∈ X, F x = F' x) → Phi X F' = Phi X F) → unpack_u_o (pack_u X F) Phi = Phi X F

Primitive. The name pack_b is a term of type set → (set → set → set) → set.

Axiom. (pack_b_0_eq) We take the following as an axiom:

∀S X, ∀F : set → set → set, S = pack_b X F → X = S 0

Axiom. (pack_b_0_eq2) We take the following as an axiom:

∀X, ∀F : set → set → set, X = pack_b X F 0

Axiom. (pack_b_1_eq) We take the following as an axiom:

∀S X, ∀F : set → set → set, S = pack_b X F → ∀x y ∈ X, F x y = decode_b (S 1) x y

Axiom. (pack_b_1_eq2) We take the following as an axiom:

∀X, ∀F : set → set → set, ∀x y ∈ X, F x y = decode_b (pack_b X F 1) x y

Axiom. (pack_b_inj) We take the following as an axiom:

∀X X', ∀F F' : set → set → set, pack_b X F = pack_b X' F' → X = X' ∧ ∀x y ∈ X, F x y = F' x y

Axiom. (pack_b_ext) We take the following as an axiom:

∀X, ∀F F' : set → set → set, (∀x y ∈ X, F x y = F' x y) → pack_b X F = pack_b X F'

Definition. We define struct_b to be λS ⇒ ∀q : set → prop, (∀X : set, ∀F : set → set → set, (∀x y ∈ X, F x y ∈ X) → q (pack_b X F)) → q S of type set → prop.

Axiom. (pack_struct_b_I) We take the following as an axiom:

∀X, ∀F : set → set → set, (∀x y ∈ X, F x y ∈ X) → struct_b (pack_b X F)

Axiom. (pack_struct_b_E1) We take the following as an axiom:

∀X, ∀F : set → set → set, struct_b (pack_b X F) → ∀x y ∈ X, F x y ∈ X

Axiom. (struct_b_eta) We take the following as an axiom:

∀S, struct_b S → S = pack_b (S 0) (decode_b (S 1))

Primitive. The name unpack_b_i is a term of type set → (set → (set → set → set) → set) → set.

Axiom. (unpack_b_i_eq) We take the following as an axiom:

∀Phi : set → (set → set → set) → set, ∀X, ∀F : set → set → set, (∀F' : set → set → set, (∀x y ∈ X, F x y = F' x y) → Phi X F' = Phi X F) → unpack_b_i (pack_b X F) Phi = Phi X F

Primitive. The name unpack_b_o is a term of type set → (set → (set → set → set) → prop) → prop.

Axiom. (unpack_b_o_eq) We take the following as an axiom:

∀Phi : set → (set → set → set) → prop, ∀X, ∀F : set → set → set, (∀F' : set → set → set, (∀x y ∈ X, F x y = F' x y) → Phi X F' = Phi X F) → unpack_b_o (pack_b X F) Phi = Phi X F

Primitive. The name pack_p is a term of type set → (set → prop) → set.

Axiom. (pack_p_0_eq) We take the following as an axiom:

∀S X, ∀P : set → prop, S = pack_p X P → X = S 0

Axiom. (pack_p_0_eq2) We take the following as an axiom:

∀X, ∀P : set → prop, X = pack_p X P 0

Axiom. (pack_p_1_eq) We take the following as an axiom:

∀S X, ∀P : set → prop, S = pack_p X P → ∀x ∈ X, P x = decode_p (S 1) x

Axiom. (pack_p_1_eq2) We take the following as an axiom:

∀X, ∀P : set → prop, ∀x ∈ X, P x = decode_p (pack_p X P 1) x

Axiom. (pack_p_inj) We take the following as an axiom:

∀X X', ∀P P' : set → prop, pack_p X P = pack_p X' P' → X = X' ∧ ∀x ∈ X, P x = P' x

Axiom. (pack_p_ext) We take the following as an axiom:

∀X, ∀P P' : set → prop, (∀x ∈ X, P x ↔ P' x) → pack_p X P = pack_p X P'

Definition. We define struct_p to be λS ⇒ ∀q : set → prop, (∀X : set, ∀P : set → prop, q (pack_p X P)) → q S of type set → prop.

Axiom. (pack_struct_p_I) We take the following as an axiom:

∀X, ∀P : set → prop, struct_p (pack_p X P)

Axiom. (struct_p_eta) We take the following as an axiom:

∀S, struct_p S → S = pack_p (S 0) (decode_p (S 1))

Primitive. The name unpack_p_i is a term of type set → (set → (set → prop) → set) → set.

Axiom. (unpack_p_i_eq) We take the following as an axiom:

∀Phi : set → (set → prop) → set, ∀X, ∀P : set → prop, (∀P' : set → prop, (∀x ∈ X, P x ↔ P' x) → Phi X P' = Phi X P) → unpack_p_i (pack_p X P) Phi = Phi X P

Primitive. The name unpack_p_o is a term of type set → (set → (set → prop) → prop) → prop.

Axiom. (unpack_p_o_eq) We take the following as an axiom:

∀Phi : set → (set → prop) → prop, ∀X, ∀P : set → prop, (∀P' : set → prop, (∀x ∈ X, P x ↔ P' x) → Phi X P' = Phi X P) → unpack_p_o (pack_p X P) Phi = Phi X P

Primitive. The name pack_r is a term of type set → (set → set → prop) → set.

Axiom. (pack_r_0_eq) We take the following as an axiom:

∀S X, ∀R : set → set → prop, S = pack_r X R → X = S 0

Axiom. (pack_r_0_eq2) We take the following as an axiom:

∀X, ∀R : set → set → prop, X = pack_r X R 0

Axiom. (pack_r_1_eq) We take the following as an axiom:

∀S X, ∀R : set → set → prop, S = pack_r X R → ∀x y ∈ X, R x y = decode_r (S 1) x y

Axiom. (pack_r_1_eq2) We take the following as an axiom:

∀X, ∀R : set → set → prop, ∀x y ∈ X, R x y = decode_r (pack_r X R 1) x y

Axiom. (pack_r_inj) We take the following as an axiom:

∀X X', ∀R R' : set → set → prop, pack_r X R = pack_r X' R' → X = X' ∧ ∀x y ∈ X, R x y = R' x y

Axiom. (pack_r_ext) We take the following as an axiom:

∀X, ∀R R' : set → set → prop, (∀x y ∈ X, R x y ↔ R' x y) → pack_r X R = pack_r X R'

Definition. We define struct_r to be λS ⇒ ∀q : set → prop, (∀X : set, ∀R : set → set → prop, q (pack_r X R)) → q S of type set → prop.

Axiom. (pack_struct_r_I) We take the following as an axiom:

∀X, ∀R : set → set → prop, struct_r (pack_r X R)

Axiom. (struct_r_eta) We take the following as an axiom:

∀S, struct_r S → S = pack_r (S 0) (decode_r (S 1))

Primitive. The name unpack_r_i is a term of type set → (set → (set → set → prop) → set) → set.

Axiom. (unpack_r_i_eq) We take the following as an axiom:

∀Phi : set → (set → set → prop) → set, ∀X, ∀R : set → set → prop, (∀R' : set → set → prop, (∀x y ∈ X, R x y ↔ R' x y) → Phi X R' = Phi X R) → unpack_r_i (pack_r X R) Phi = Phi X R

Primitive. The name unpack_r_o is a term of type set → (set → (set → set → prop) → prop) → prop.

Axiom. (unpack_r_o_eq) We take the following as an axiom:

∀Phi : set → (set → set → prop) → prop, ∀X, ∀R : set → set → prop, (∀R' : set → set → prop, (∀x y ∈ X, R x y ↔ R' x y) → Phi X R' = Phi X R) → unpack_r_o (pack_r X R) Phi = Phi X R

Primitive. The name pack_c is a term of type set → ((set → prop) → prop) → set.

Axiom. (pack_c_0_eq) We take the following as an axiom:

∀S X, ∀C : (set → prop) → prop, S = pack_c X C → X = S 0

Axiom. (pack_c_0_eq2) We take the following as an axiom:

∀X, ∀C : (set → prop) → prop, X = pack_c X C 0

Axiom. (pack_c_1_eq) We take the following as an axiom:

∀S X, ∀C : (set → prop) → prop, S = pack_c X C → ∀U : set → prop, (∀x, U x → x ∈ X) → C U = decode_c (S 1) U

Axiom. (pack_c_1_eq2) We take the following as an axiom:

∀X, ∀C : (set → prop) → prop, ∀U : set → prop, (∀x, U x → x ∈ X) → C U = decode_c (pack_c X C 1) U

Axiom. (pack_c_inj) We take the following as an axiom:

∀X X', ∀C C' : (set → prop) → prop, pack_c X C = pack_c X' C' → X = X' ∧ ∀U : set → prop, (∀x, U x → x ∈ X) → C U = C' U

Axiom. (pack_c_ext) We take the following as an axiom:

∀X, ∀C C' : (set → prop) → prop, (∀U : set → prop, (∀x, U x → x ∈ X) → (C U ↔ C' U)) → pack_c X C = pack_c X C'

Definition. We define struct_c to be λS ⇒ ∀q : set → prop, (∀X : set, ∀C : (set → prop) → prop, q (pack_c X C)) → q S of type set → prop.

Axiom. (pack_struct_c_I) We take the following as an axiom:

∀X, ∀C : (set → prop) → prop, struct_c (pack_c X C)

Axiom. (struct_c_eta) We take the following as an axiom:

∀S, struct_c S → S = pack_c (S 0) (decode_c (S 1))

Primitive. The name unpack_c_i is a term of type set → (set → ((set → prop) → prop) → set) → set.

Axiom. (unpack_c_i_eq) We take the following as an axiom:

∀Phi : set → ((set → prop) → prop) → set, ∀X, ∀C : (set → prop) → prop, (∀C' : (set → prop) → prop, (∀U : set → prop, (∀x, U x → x ∈ X) → (C U ↔ C' U)) → Phi X C' = Phi X C) → unpack_c_i (pack_c X C) Phi = Phi X C

Primitive. The name unpack_c_o is a term of type set → (set → ((set → prop) → prop) → prop) → prop.

Axiom. (unpack_c_o_eq) We take the following as an axiom:

∀Phi : set → ((set → prop) → prop) → prop, ∀X, ∀C : (set → prop) → prop, (∀C' : (set → prop) → prop, (∀U : set → prop, (∀x, U x → x ∈ X) → (C U ↔ C' U)) → Phi X C' = Phi X C) → unpack_c_o (pack_c X C) Phi = Phi X C

Primitive. The name canonical_elt is a term of type (set → set → prop) → set → set.

Axiom. (canonical_elt_rel) We take the following as an axiom:

∀R : set → set → prop, ∀x : set, R x x → R x (canonical_elt R x)

Axiom. (canonical_elt_eq) We take the following as an axiom:

∀R : set → set → prop, per R → ∀x y : set, R x y → canonical_elt R x = canonical_elt R y

Axiom. (canonical_elt_idem) We take the following as an axiom:

∀R : set → set → prop, per R → ∀x : set, R x x → canonical_elt R x = canonical_elt R (canonical_elt R x)

Primitive. The name quotient is a term of type (set → set → prop) → set → prop.

Axiom. (quotient_prop1) We take the following as an axiom:

∀R : set → set → prop, ∀x : set, quotient R x → R x x

Axiom. (quotient_prop2) We take the following as an axiom:

∀R : set → set → prop, per R → ∀x y : set, quotient R x → quotient R y → R x y → x = y

Primitive. The name canonical_elt_def is a term of type (set → set → prop) → (set → set) → set → set.

Axiom. (canonical_elt_def_rel) We take the following as an axiom:

∀R : set → set → prop, ∀d : set → set, ∀x : set, R x x → R x (canonical_elt_def R d x)

Axiom. (canonical_elt_def_eq) We take the following as an axiom:

∀R : set → set → prop, per R → ∀d : set → set, (∀x y : set, R x y → d x = d y) → ∀x y : set, R x y → canonical_elt_def R d x = canonical_elt_def R d y

Axiom. (canonical_elt_def_idem) We take the following as an axiom:

∀R : set → set → prop, per R → ∀d : set → set, (∀x y : set, R x y → d x = d y) → ∀x : set, R x x → canonical_elt_def R d x = canonical_elt_def R d (canonical_elt_def R d x)

Primitive. The name quotient_def is a term of type (set → set → prop) → (set → set) → set → prop.

Axiom. (quotient_def_prop0) We take the following as an axiom:

∀R : set → set → prop, per R → ∀d : set → set, ∀x : set, R x (d x) → x = d x → quotient_def R d x

Axiom. (quotient_def_prop1) We take the following as an axiom:

∀R : set → set → prop, ∀d : set → set, ∀x : set, quotient_def R d x → R x x

Axiom. (quotient_def_prop2) We take the following as an axiom:

∀R : set → set → prop, per R → ∀d : set → set, (∀x y : set, R x y → d x = d y) → ∀x y : set, quotient_def R d x → quotient_def R d y → R x y → x = y

Beginning of Section explicit_Nats

Variable N : set

Variable base : set

Variable S : set → set

Primitive. The name explicit_Nats is a term of type prop.

Axiom. (explicit_Nats_I) We take the following as an axiom:

(base ∈ N) → (∀m ∈ N, S m ∈ N) → (∀m ∈ N, S m ≠ base) → (∀m n ∈ N, S m = S n → m = n) → (∀p : set → prop, p base → (∀m, p m → p (S m)) → (∀m ∈ N, p m)) → explicit_Nats

Axiom. (explicit_Nats_E) We take the following as an axiom:

∀q : prop, (explicit_Nats → (base ∈ N) → (∀m ∈ N, S m ∈ N) → (∀m ∈ N, S m ≠ base) → (∀m n ∈ N, S m = S n → m = n) → (∀p : set → prop, p base → (∀m, p m → p (S m)) → (∀m ∈ N, p m)) → q) → explicit_Nats → q

Axiom. (explicit_Nats_ind) We take the following as an axiom:

explicit_Nats → ∀p : set → prop, p base → (∀m ∈ N, p m → p (S m)) → ∀m ∈ N, p m

Primitive. The name explicit_Nats_primrec is a term of type set → (set → set → set) → set → set.

Axiom. (explicit_Nats_primrec_base) We take the following as an axiom:

∀a, ∀f : set → set → set, explicit_Nats → explicit_Nats_primrec a f base = a

Axiom. (explicit_Nats_primrec_S) We take the following as an axiom:

∀a, ∀f : set → set → set, explicit_Nats → ∀n ∈ N, explicit_Nats_primrec a f (S n) = f n (explicit_Nats_primrec a f n)

Axiom. (explicit_Nats_primrec_P) We take the following as an axiom:

explicit_Nats → ∀P : set → prop, ∀a, P a → ∀f : set → set → set, (∀n ∈ N, ∀b, P b → P (f n b)) → ∀n ∈ N, P (explicit_Nats_primrec a f n)

End of Section explicit_Nats

Axiom. (explicit_Nats_omega) We take the following as an axiom:

explicit_Nats ω 0 ordsucc

Beginning of Section explicit_Nats_zero

Variable N : set

Variable zero : set

Variable S : set → set

Primitive. The name explicit_Nats_zero_plus is a term of type set → set → set.

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term explicit_Nats_zero_plus.

Primitive. The name explicit_Nats_zero_mult is a term of type set → set → set.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term explicit_Nats_zero_mult.

Hypothesis HN : explicit_Nats N zero S

Axiom. (explicit_Nats_zero_plus_N) We take the following as an axiom:

∀n m ∈ N, n + m ∈ N

Axiom. (explicit_Nats_zero_plus_0L) We take the following as an axiom:

∀m ∈ N, zero + m = m

Axiom. (explicit_Nats_zero_plus_SL) We take the following as an axiom:

∀n m ∈ N, S n + m = S (n + m)

Axiom. (explicit_Nats_zero_mult_N) We take the following as an axiom:

∀n m ∈ N, n * m ∈ N

Axiom. (explicit_Nats_zero_mult_0L) We take the following as an axiom:

∀m ∈ N, zero * m = zero

Axiom. (explicit_Nats_zero_mult_SL) We take the following as an axiom:

∀n m ∈ N, S n * m = m + n * m

End of Section explicit_Nats_zero

Beginning of Section explicit_Nats_one

Variable N : set

Variable one : set

Variable S : set → set

Primitive. The name explicit_Nats_one_plus is a term of type set → set → set.

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term explicit_Nats_one_plus.

Primitive. The name explicit_Nats_one_mult is a term of type set → set → set.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term explicit_Nats_one_mult.

Primitive. The name explicit_Nats_one_exp is a term of type set → set → set.

Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term explicit_Nats_one_exp.

Hypothesis HN : explicit_Nats N one S

Axiom. (explicit_Nats_one_plus_N) We take the following as an axiom:

∀n m ∈ N, n + m ∈ N

Axiom. (explicit_Nats_one_plus_1L) We take the following as an axiom:

∀m ∈ N, one + m = S m

Axiom. (explicit_Nats_one_plus_SL) We take the following as an axiom:

∀n m ∈ N, S n + m = S (n + m)

Axiom. (explicit_Nats_one_mult_N) We take the following as an axiom:

∀n m ∈ N, n * m ∈ N

Axiom. (explicit_Nats_one_mult_1L) We take the following as an axiom:

∀m ∈ N, one * m = m

Axiom. (explicit_Nats_one_mult_SL) We take the following as an axiom:

∀n m ∈ N, S n * m = m + n * m

Axiom. (explicit_Nats_one_exp_N) We take the following as an axiom:

∀n m ∈ N, n ^ m ∈ N

Axiom. (explicit_Nats_one_exp_1L) We take the following as an axiom:

∀n ∈ N, n ^ one = n

Axiom. (explicit_Nats_one_exp_SL) We take the following as an axiom:

∀n m ∈ N, n ^ (S m) = n * n ^ m

Definition. We define explicit_Nats_one_lt to be λm n ⇒ m ∈ N ∧ n ∈ N ∧ ∃k ∈ N, m + k = n of type set → set → prop.

Definition. We define explicit_Nats_one_le to be λm n ⇒ m ∈ N ∧ n ∈ N ∧ (m = n ∨ ∃k ∈ N, m + k = n) of type set → set → prop.

Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term explicit_Nats_one_lt.

Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term explicit_Nats_one_le.

End of Section explicit_Nats_one

Beginning of Section explicit_Nats_transfer

Variable N : set

Variable base : set

Variable S : set → set

Variable N' : set

Variable base' : set

Variable S' : set → set

Variable f : set → set

Axiom. (explicit_Nats_transfer) We take the following as an axiom:

explicit_Nats N base S → bij N N' f → f base = base' → (∀n ∈ N, f (S n) = S' (f n)) → explicit_Nats N' base' S'

End of Section explicit_Nats_transfer

Beginning of Section AssocComm

Variable R : set

Variable plus : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Axiom. (AssocComm_identities) We take the following as an axiom:

(∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → ∀p : prop, ((∀x y z ∈ R, x + y + z = y + x + z) → (∀x y z ∈ R, x + y + z = z + x + y) → (∀x y z w ∈ R, (x + y) + (z + w) = (x + z) + (y + w)) → (∀x y z w ∈ R, x + y + z + w = w + x + y + z) → (∀x y z w ∈ R, x + y + z + w = z + w + x + y) → p) → p

End of Section AssocComm

Beginning of Section Group1

Variable G : set

Beginning of Section Group1Explicit

Variable op : set → set → set

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term op.

Definition. We define explicit_Group to be (∀a b ∈ G, a * b ∈ G) ∧ (∀a b c ∈ G, a * (b * c) = (a * b) * c) ∧ ∃e ∈ G, (∀a ∈ G, e * a = a ∧ a * e = a) ∧ (∀a ∈ G, ∃b ∈ G, a * b = e ∧ b * a = e) of type prop.

Axiom. (explicit_Group_identity_unique) We take the following as an axiom:

∀e e' ∈ G, (∀a ∈ G, e * a = a) → (∀a ∈ G, a * e' = a) → e = e'

Hypothesis HG : explicit_Group

Definition. We define explicit_Group_identity to be Eps_i (λe ⇒ e ∈ G ∧ ((∀a ∈ G, e * a = a ∧ a * e = a) ∧ ∀a ∈ G, ∃b ∈ G, a * b = e ∧ b * a = e)) of type set.

Let e ≝ explicit_Group_identity

Definition. We define explicit_Group_inverse to be λa ⇒ Eps_i (λb ⇒ b ∈ G ∧ (a * b = e ∧ b * a = e)) of type set → set.

Notation. We use - as a postfix operator with priority 340 corresponding to applying term explicit_Group_inverse.

Axiom. (explicit_Group_identity_prop) We take the following as an axiom:

e ∈ G ∧ ((∀a ∈ G, e * a = a ∧ a * e = a) ∧ ∀a ∈ G, ∃b ∈ G, a * b = e ∧ b * a = e)

Axiom. (explicit_Group_identity_in) We take the following as an axiom:

e ∈ G

Axiom. (explicit_Group_identity_lid) We take the following as an axiom:

∀a ∈ G, e * a = a

Axiom. (explicit_Group_identity_rid) We take the following as an axiom:

∀a ∈ G, a * e = a

Axiom. (explicit_Group_identity_invex) We take the following as an axiom:

∀a ∈ G, ∃b ∈ G, a * b = e ∧ b * a = e

Axiom. (explicit_Group_inverse_prop) We take the following as an axiom:

∀a ∈ G, a - ∈ G ∧ (a * a - = e ∧ a - * a = e)

Axiom. (explicit_Group_inverse_in) We take the following as an axiom:

∀a ∈ G, a - ∈ G

Axiom. (explicit_Group_inverse_rinv) We take the following as an axiom:

∀a ∈ G, a * a - = e

Axiom. (explicit_Group_inverse_linv) We take the following as an axiom:

∀a ∈ G, a - * a = e

Axiom. (explicit_Group_lcancel) We take the following as an axiom:

∀a b c ∈ G, a * b = a * c → b = c

Axiom. (explicit_Group_rcancel) We take the following as an axiom:

∀a b c ∈ G, a * c = b * c → a = b

Axiom. (explicit_Group_rinv_rev) We take the following as an axiom:

∀a b ∈ G, a * b = e → b = a -

Axiom. (explicit_Group_inv_com) We take the following as an axiom:

∀a b ∈ G, a * b = e → b * a = e

Axiom. (explicit_Group_inv_rev2) We take the following as an axiom:

∀a b ∈ G, (a * b) * (a * b) = e → (b * a) * (b * a) = e

Definition. We define explicit_abelian to be ∀a b ∈ G, a * b = b * a of type prop.

End of Section Group1Explicit

Beginning of Section Group1Explicit2

Variable op : set → set → set

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term op.

Beginning of Section Group1Explicit2RepIndep

Variable op' : set → set → set

Notation. We use ⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term op'.

Hypothesis Hopop' : ∀a b ∈ G, a * b = a ⨯ b

Axiom. (explicit_Group_repindep_imp) We take the following as an axiom:

explicit_Group op → explicit_Group op'

Let e ≝ explicit_Group_identity op

Let e' ≝ explicit_Group_identity op'

Axiom. (explicit_Group_identity_repindep) We take the following as an axiom:

explicit_Group op → e = e'

Let inv ≝ explicit_Group_inverse op

Let inv' ≝ explicit_Group_inverse op'

Axiom. (explicit_Group_inverse_repindep) We take the following as an axiom:

explicit_Group op → ∀a ∈ G, inv a = inv' a

Axiom. (explicit_abelian_repindep_imp) We take the following as an axiom:

explicit_abelian op → explicit_abelian op'

End of Section Group1Explicit2RepIndep

End of Section Group1Explicit2

Beginning of Section Group1Explicit3RepIndep

Variable op : set → set → set

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term op.

Variable op' : set → set → set

Notation. We use ⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term op'.

Hypothesis Hopop' : ∀a b ∈ G, a * b = a ⨯ b

Axiom. (explicit_Group_repindep) We take the following as an axiom:

explicit_Group op ↔ explicit_Group op'

Axiom. (explicit_abelian_repindep) We take the following as an axiom:

explicit_abelian op ↔ explicit_abelian op'

End of Section Group1Explicit3RepIndep

End of Section Group1

Definition. We define Group to be λG ⇒ struct_b G ∧ unpack_b_o G explicit_Group of type set → prop.

Definition. We define abelian_Group to be λG ⇒ Group G ∧ unpack_b_o G explicit_abelian of type set → prop.

Axiom. (Group_unpack_eq) We take the following as an axiom:

∀G, ∀op : set → set → set, unpack_b_o (pack_b G op) explicit_Group = explicit_Group G op

Axiom. (GroupI) We take the following as an axiom:

∀G, ∀op : set → set → set, explicit_Group G op → Group (pack_b G op)

Axiom. (GroupE) We take the following as an axiom:

∀G, ∀op : set → set → set, Group (pack_b G op) → explicit_Group G op

Axiom. (abelian_Group_unpack_eq) We take the following as an axiom:

∀G, ∀op : set → set → set, unpack_b_o (pack_b G op) explicit_abelian = explicit_abelian G op

Axiom. (abelian_Group_E) We take the following as an axiom:

∀G, ∀op : set → set → set, abelian_Group (pack_b G op) → Group (pack_b G op) ∧ explicit_abelian G op

Beginning of Section Group2

Variable G : set

Variable op : set → set → set

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term op.

Notation. We use - as a postfix operator with priority 340 corresponding to applying term explicit_Group_inverse G op.

Variable H : set

Definition. We define explicit_subgroup to be Group (pack_b H op) ∧ H ⊆ G of type prop.

Definition. We define explicit_normal to be ∀x ∈ G, {x * a * x -|a ∈ H} ⊆ H of type prop.

Hypothesis HG : Group (pack_b G op)

Let e ≝ explicit_Group_identity G op

Axiom. (explicit_subgroup_test) We take the following as an axiom:

H ⊆ G → e ∈ H → (∀a ∈ H, a - ∈ H) → (∀a b ∈ H, a * b ∈ H) → explicit_subgroup

Hypothesis HSG : explicit_subgroup

Let e' ≝ explicit_Group_identity H op

Axiom. (explicit_subgroup_identity_eq) We take the following as an axiom:

e = e'

Axiom. (explicit_subgroup_inv_eq) We take the following as an axiom:

∀a ∈ H, explicit_Group_inverse G op a = explicit_Group_inverse H op a

Axiom. (explicit_abelian_normal) We take the following as an axiom:

explicit_abelian G op → explicit_normal

End of Section Group2

Beginning of Section Group3

Variable H G : set

Variable op op' : set → set → set

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term op.

Notation. We use - as a postfix operator with priority 340 corresponding to applying term explicit_Group_inverse G op.

Notation. We use ⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term op'.

Notation. We use :-: as a postfix operator with priority 340 corresponding to applying term explicit_Group_inverse G op'.

Hypothesis HG : explicit_Group G op

Hypothesis HHG : H ⊆ G

Hypothesis Hopop' : ∀a b ∈ G, a * b = a ⨯ b

Axiom. (explicit_normal_repindep_imp) We take the following as an axiom:

explicit_normal G op H → explicit_normal G op' H

End of Section Group3

Definition. We define subgroup to be λH G ⇒ struct_b G ∧ struct_b H ∧ unpack_b_o G (λG' op ⇒ unpack_b_o H (λH' _ ⇒ H = pack_b H' op ∧ Group (pack_b H' op) ∧ H' ⊆ G')) of type set → set → prop.

Notation. We use ≤ as an infix operator with priority 400 and no associativity corresponding to applying term subgroup.

Definition. We define subgroup_index to be λH G ⇒ unpack_b_i G (λG' op ⇒ {n ∈ ω|∃f ∈ G'^{ordsucc n}, ∀i j ∈ ordsucc n, i ≠ j → ∀a b ∈ H 0, op (f i) a ≠ op (f j) b}) of type set → set → set.

Definition. We define normal_subgroup to be λH G ⇒ H ≤ G ∧ unpack_b_o G (λG' op ⇒ unpack_b_o H (λH' _ ⇒ explicit_normal G' op H')) of type set → set → prop.

Axiom. (pack_b_subgroup_E) We take the following as an axiom:

∀H G : set, ∀opH op : set → set → set, pack_b H opH ≤ pack_b G op → pack_b H opH = pack_b H op ∧ explicit_subgroup G op H

Axiom. (subgroup_E) We take the following as an axiom:

∀H G, H ≤ G → ∀q : set → set → prop, (∀H G, ∀op : set → set → set, (∀a b ∈ G, op a b ∈ G) → Group (pack_b H op) → H ⊆ G → q (pack_b H op) (pack_b G op)) → q H G

Axiom. (abelian_group_normal_subgroup) We take the following as an axiom:

∀G, abelian_Group G → ∀H, H ≤ G → normal_subgroup H G

Axiom. (subgroup_transitive) We take the following as an axiom:

∀K H G, K ≤ H → H ≤ G → K ≤ G

Beginning of Section Group4

Variable A : set

Let G ≝ {f ∈ A^A|bij A A (λx ⇒ f x)}

Let op ≝ λf g : set ⇒ λx ∈ A ⇒ g (f x)

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term op.

Let id ≝ λx ∈ A ⇒ x

Axiom. (explicit_Group_symgroup) We take the following as an axiom:

explicit_Group G op

Axiom. (explicit_Group_symgroup_id_eq) We take the following as an axiom:

explicit_Group_identity G op = id

Axiom. (explicit_Group_symgroup_inv_eq) We take the following as an axiom:

∀f ∈ G, explicit_Group_inverse G op f = (λx ∈ A ⇒ inv A (λx ⇒ f x) x)

Variable B : set

Let H ≝ {f ∈ A^A|bij A A (λx ⇒ f x) ∧ ∀x ∈ B, f x = x}

Axiom. (explicit_subgroup_symgroup_fixing) We take the following as an axiom:

B ⊆ A → explicit_subgroup G op H

End of Section Group4

Definition. We define symgroup to be λA ⇒ pack_b {f ∈ A^A|bij A A (λx ⇒ f x)} (λf g ⇒ λx ∈ A ⇒ g (f x)) of type set → set.

Definition. We define symgroup_fixing to be λA B ⇒ pack_b {f ∈ A^A|bij A A (λx ⇒ f x) ∧ ∀x ∈ B, f x = x} (λf g ⇒ λx ∈ A ⇒ g (f x)) of type set → set → set.

Axiom. (Group_symgroup) We take the following as an axiom:

∀A, Group (symgroup A)

Axiom. (Group_symgroup_fixing) We take the following as an axiom:

∀A B, B ⊆ A → Group (symgroup_fixing A B)

Axiom. (subgroup_symgroup_fixing) We take the following as an axiom:

∀A B, B ⊆ A → symgroup_fixing A B ≤ symgroup A

Axiom. (subgroup_symgroup_fixing2) We take the following as an axiom:

∀A B C, C ⊆ B → B ⊆ A → symgroup_fixing A B ≤ symgroup_fixing A C

Axiom. (nonnormal_subgroup) We take the following as an axiom:

∃H G, Group G ∧ H ≤ G ∧ ¬ normal_subgroup H G

Definition. We define normal_subgroup_equiv to be λG N a b ⇒ unpack_b_o G (λG op ⇒ a ∈ G ∧ b ∈ G ∧ op a (explicit_Group_inverse G op b) ∈ N 0) of type set → set → set → set → prop.

Definition. We define quotient_Group to be λG N ⇒ unpack_b_i G (λG' op ⇒ pack_b {a ∈ G'|quotient (normal_subgroup_equiv G N) a} (λa b ⇒ canonical_elt (normal_subgroup_equiv G N) (op a b))) of type set → set → set.

Definition. We define trivial_Group_p to be λG ⇒ Group G ∧ ∀x y ∈ G 0, x = y of type set → prop.

Definition. We define solvable_Group_p to be λG ⇒ ∃n ∈ ω, ∃Gseq, (∀i ∈ ordsucc n, Group (Gseq i)) ∧ (∀i ∈ n, normal_subgroup (Gseq (ordsucc i)) (Gseq i)) ∧ (∀i ∈ n, abelian_Group (quotient_Group (Gseq i) (Gseq (ordsucc i)))) ∧ G = Gseq 0 ∧ trivial_Group_p (Gseq n) of type set → prop.

Definition. We define Group_carrier to be λGs ⇒ Gs 0 of type set → set.

Definition. We define Group_op to be λGs ⇒ decode_b (Gs 1) of type set → set → set → set.

Beginning of Section Group2

Variable Gs : set

Variable Gs' : set

Let G : set ≝ Group_carrier Gs

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term Group_op Gs.

Let G' : set ≝ Group_carrier Gs'

Notation. We use ⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term Group_op Gs'.

Definition. We define Group_Hom to be λg ⇒ Group Gs ∧ Group Gs' ∧ g ∈ G'^G ∧ ∀a b ∈ G, g (a * b) = g a ⨯ g b of type set → prop.

Definition. We define Group_Iso to be λg ⇒ Group_Hom g ∧ bij G G' (λx ⇒ g x) of type set → prop.

Definition. We define Group_Isomorphic to be ∃g, Group_Iso g of type prop.

End of Section Group2

Beginning of Section explicit_Ring

Variable R : set

Variable zero : set

Variable plus mult : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Primitive. The name explicit_Ring is a term of type prop.

Axiom. (explicit_Ring_I) We take the following as an axiom:

(∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → zero ∈ R → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → (∀x y z ∈ R, (x + y) * z = x * z + y * z) → explicit_Ring

Axiom. (explicit_Ring_E) We take the following as an axiom:

∀q : prop, (explicit_Ring → (∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → (zero ∈ R) → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → (∀x y z ∈ R, (x + y) * z = x * z + y * z) → q) → explicit_Ring → q

Primitive. The name explicit_Ring_minus is a term of type set → set.

Notation. We use - as a prefix operator with priority 358 corresponding to applying term explicit_Ring_minus.

Axiom. (explicit_Ring_minus_prop) We take the following as an axiom:

explicit_Ring → ∀x ∈ R, - x ∈ R ∧ x + - x = zero

Axiom. (explicit_Ring_minus_clos) We take the following as an axiom:

explicit_Ring → ∀x ∈ R, - x ∈ R

Axiom. (explicit_Ring_minus_R) We take the following as an axiom:

explicit_Ring → ∀x ∈ R, x + - x = zero

Axiom. (explicit_Ring_minus_L) We take the following as an axiom:

explicit_Ring → ∀x ∈ R, - x + x = zero

Axiom. (explicit_Ring_plus_cancelL) We take the following as an axiom:

explicit_Ring → ∀x y z ∈ R, x + y = x + z → y = z

Axiom. (explicit_Ring_plus_cancelR) We take the following as an axiom:

explicit_Ring → ∀x y z ∈ R, x + z = y + z → x = y

Axiom. (explicit_Ring_minus_invol) We take the following as an axiom:

explicit_Ring → ∀x ∈ R, - - x = x

End of Section explicit_Ring

Beginning of Section explicit_Ring_with_id

Variable R : set

Variable zero one : set

Variable plus mult : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Primitive. The name explicit_Ring_with_id is a term of type prop.

Axiom. (explicit_Ring_with_id_I) We take the following as an axiom:

(∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → zero ∈ R → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R, one * x = x) → (∀x ∈ R, x * one = x) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → (∀x y z ∈ R, (x + y) * z = x * z + y * z) → explicit_Ring_with_id

Axiom. (explicit_Ring_with_id_E) We take the following as an axiom:

∀q : prop, (explicit_Ring_with_id → (∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → (zero ∈ R) → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R, one * x = x) → (∀x ∈ R, x * one = x) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → (∀x y z ∈ R, (x + y) * z = x * z + y * z) → q) → explicit_Ring_with_id → q

Axiom. (explicit_Ring_with_id_Ring) We take the following as an axiom:

explicit_Ring_with_id → explicit_Ring R zero plus mult

Notation. We use - as a prefix operator with priority 358 corresponding to applying term explicit_Ring_minus R zero plus mult.

Axiom. (explicit_Ring_with_id_minus_clos) We take the following as an axiom:

explicit_Ring_with_id → ∀x ∈ R, - x ∈ R

Axiom. (explicit_Ring_with_id_minus_R) We take the following as an axiom:

explicit_Ring_with_id → ∀x ∈ R, x + - x = zero

Axiom. (explicit_Ring_with_id_minus_L) We take the following as an axiom:

explicit_Ring_with_id → ∀x ∈ R, - x + x = zero

Axiom. (explicit_Ring_with_id_plus_cancelL) We take the following as an axiom:

explicit_Ring_with_id → ∀x y z ∈ R, x + y = x + z → y = z

Axiom. (explicit_Ring_with_id_plus_cancelR) We take the following as an axiom:

explicit_Ring_with_id → ∀x y z ∈ R, x + z = y + z → x = y

Axiom. (explicit_Ring_with_id_minus_invol) We take the following as an axiom:

explicit_Ring_with_id → ∀x ∈ R, - - x = x

Axiom. (explicit_Ring_with_id_minus_one_In) We take the following as an axiom:

explicit_Ring_with_id → - one ∈ R

Axiom. (explicit_Ring_with_id_zero_multR) We take the following as an axiom:

explicit_Ring_with_id → ∀x ∈ R, x * zero = zero

Axiom. (explicit_Ring_with_id_zero_multL) We take the following as an axiom:

explicit_Ring_with_id → ∀x ∈ R, zero * x = zero

Axiom. (explicit_Ring_with_id_minus_mult) We take the following as an axiom:

explicit_Ring_with_id → ∀x ∈ R, - x = (- one) * x

Axiom. (explicit_Ring_with_id_mult_minus) We take the following as an axiom:

explicit_Ring_with_id → ∀x ∈ R, - x = x * (- one)

Axiom. (explicit_Ring_with_id_minus_one_square) We take the following as an axiom:

explicit_Ring_with_id → (- one) * (- one) = one

Axiom. (explicit_Ring_with_id_minus_square) We take the following as an axiom:

explicit_Ring_with_id → ∀x ∈ R, (- x) * (- x) = x * x

Definition. We define explicit_Ring_with_id_exp_nat to be λx n ⇒ nat_primrec one (λ_ r ⇒ x * r) n of type set → set → set.

Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term explicit_Ring_with_id_exp_nat.

Definition. We define explicit_Ring_with_id_eval_poly to be λn cs x ⇒ nat_primrec zero (λm r ⇒ cs m * x ^ m + r) n of type set → set → set → set.

End of Section explicit_Ring_with_id

Beginning of Section explicit_Ring_with_id_RepIndep2

Variable R : set

Variable zero one : set

Variable plus mult : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Variable plus' mult' : set → set → set

Notation. We use + as an infix operator with priority 355 and which associates to the right corresponding to applying term plus'.

Notation. We use ⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term mult'.

Hypothesis Hpp' : ∀a b ∈ R, a + b = a + b

Hypothesis Hmm' : ∀a b ∈ R, a * b = a ⨯ b

Axiom. (explicit_Ring_with_id_repindep) We take the following as an axiom:

explicit_Ring_with_id R zero one plus mult ↔ explicit_Ring_with_id R zero one plus' mult'

End of Section explicit_Ring_with_id_RepIndep2

Beginning of Section explicit_CRing_with_id

Variable R : set

Variable zero one : set

Variable plus mult : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Primitive. The name explicit_CRing_with_id is a term of type prop.

Axiom. (explicit_CRing_with_id_I) We take the following as an axiom:

(∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → zero ∈ R → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (∀x y ∈ R, x * y = y * x) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R, one * x = x) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → explicit_CRing_with_id

Axiom. (explicit_CRing_with_id_E) We take the following as an axiom:

∀q : prop, (explicit_CRing_with_id → (∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → (zero ∈ R) → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (∀x y ∈ R, x * y = y * x) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R, one * x = x) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → q) → explicit_CRing_with_id → q

Axiom. (explicit_CRing_with_id_Ring_with_id) We take the following as an axiom:

explicit_CRing_with_id → explicit_Ring_with_id R zero one plus mult

Axiom. (explicit_CRing_with_id_Ring) We take the following as an axiom:

explicit_CRing_with_id → explicit_Ring R zero plus mult

Notation. We use - as a prefix operator with priority 358 corresponding to applying term explicit_Ring_minus R zero plus mult.

Axiom. (explicit_CRing_with_id_minus_clos) We take the following as an axiom:

explicit_CRing_with_id → ∀x ∈ R, - x ∈ R

Axiom. (explicit_CRing_with_id_minus_R) We take the following as an axiom:

explicit_CRing_with_id → ∀x ∈ R, x + - x = zero

Axiom. (explicit_CRing_with_id_minus_L) We take the following as an axiom:

explicit_CRing_with_id → ∀x ∈ R, - x + x = zero

Axiom. (explicit_CRing_with_id_plus_cancelL) We take the following as an axiom:

explicit_CRing_with_id → ∀x y z ∈ R, x + y = x + z → y = z

Axiom. (explicit_CRing_with_id_plus_cancelR) We take the following as an axiom:

explicit_CRing_with_id → ∀x y z ∈ R, x + z = y + z → x = y

Axiom. (explicit_CRing_with_id_minus_invol) We take the following as an axiom:

explicit_CRing_with_id → ∀x ∈ R, - - x = x

Axiom. (explicit_CRing_with_id_minus_one_In) We take the following as an axiom:

explicit_CRing_with_id → - one ∈ R

Axiom. (explicit_CRing_with_id_zero_multR) We take the following as an axiom:

explicit_CRing_with_id → ∀x ∈ R, x * zero = zero

Axiom. (explicit_CRing_with_id_zero_multL) We take the following as an axiom:

explicit_CRing_with_id → ∀x ∈ R, zero * x = zero

Axiom. (explicit_CRing_with_id_minus_mult) We take the following as an axiom:

explicit_CRing_with_id → ∀x ∈ R, - x = (- one) * x

Axiom. (explicit_CRing_with_id_mult_minus) We take the following as an axiom:

explicit_CRing_with_id → ∀x ∈ R, - x = x * (- one)

Axiom. (explicit_CRing_with_id_minus_one_square) We take the following as an axiom:

explicit_CRing_with_id → (- one) * (- one) = one

Axiom. (explicit_CRing_with_id_minus_square) We take the following as an axiom:

explicit_CRing_with_id → ∀x ∈ R, (- x) * (- x) = x * x

End of Section explicit_CRing_with_id

Beginning of Section explicit_CRing_with_id_RepIndep2

Variable R : set

Variable zero one : set

Variable plus mult : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Variable plus' mult' : set → set → set

Notation. We use + as an infix operator with priority 355 and which associates to the right corresponding to applying term plus'.

Notation. We use ⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term mult'.

Hypothesis Hpp' : ∀a b ∈ R, a + b = a + b

Hypothesis Hmm' : ∀a b ∈ R, a * b = a ⨯ b

Axiom. (explicit_CRing_with_id_repindep) We take the following as an axiom:

explicit_CRing_with_id R zero one plus mult ↔ explicit_CRing_with_id R zero one plus' mult'

End of Section explicit_CRing_with_id_RepIndep2

Primitive. The name pack_b_b_e is a term of type set → (set → set → set) → (set → set → set) → set → set.

Primitive. The name struct_b_b_e is a term of type set → prop.

Primitive. The name unpack_b_b_e_i is a term of type set → (set → (set → set → set) → (set → set → set) → set → set) → set.

Primitive. The name unpack_b_b_e_o is a term of type set → (set → (set → set → set) → (set → set → set) → set → prop) → prop.

Definition. We define Ring to be λR ⇒ struct_b_b_e R ∧ unpack_b_b_e_o R (λR plus mult zero ⇒ explicit_Ring R zero plus mult) of type set → prop.

Definition. We define Ring_minus to be λR x ⇒ unpack_b_b_e_i R (λR plus mult zero ⇒ explicit_Ring_minus R zero plus mult x) of type set → set → set.

Primitive. The name pack_b_b_e_e is a term of type set → (set → set → set) → (set → set → set) → set → set → set.

Primitive. The name struct_b_b_e_e is a term of type set → prop.

Primitive. The name unpack_b_b_e_e_i is a term of type set → (set → (set → set → set) → (set → set → set) → set → set → set) → set.

Primitive. The name unpack_b_b_e_e_o is a term of type set → (set → (set → set → set) → (set → set → set) → set → set → prop) → prop.

Definition. We define Ring_with_id to be λR ⇒ struct_b_b_e_e R ∧ unpack_b_b_e_e_o R (λR plus mult zero one ⇒ explicit_Ring_with_id R zero one plus mult) of type set → prop.

Axiom. (Ring_with_id_unpack_eq) We take the following as an axiom:

∀R, ∀plus mult : set → set → set, ∀zero one, unpack_b_b_e_e_o (pack_b_b_e_e R plus mult zero one) (λR plus mult zero one ⇒ explicit_Ring_with_id R zero one plus mult) = explicit_Ring_with_id R zero one plus mult

Definition. We define CRing_with_id to be λR ⇒ struct_b_b_e_e R ∧ unpack_b_b_e_e_o R (λR plus mult zero one ⇒ explicit_CRing_with_id R zero one plus mult) of type set → prop.

Axiom. (CRing_with_id_unpack_eq) We take the following as an axiom:

∀R, ∀plus mult : set → set → set, ∀zero one, unpack_b_b_e_e_o (pack_b_b_e_e R plus mult zero one) (λR plus mult zero one ⇒ explicit_CRing_with_id R zero one plus mult) = explicit_CRing_with_id R zero one plus mult

Definition. We define Ring_of_Ring_with_id to be λR ⇒ unpack_b_b_e_e_i R (λR plus mult zero one ⇒ pack_b_b_e R plus mult zero) of type set → set.

Axiom. (CRing_with_id_is_Ring_with_id) We take the following as an axiom:

∀R, CRing_with_id R → Ring_with_id R

Beginning of Section explicit_Reals

Variable R : set

Variable zero one : set

Variable plus mult : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Primitive. The name explicit_Field is a term of type prop.

Axiom. (explicit_Field_I) We take the following as an axiom:

(∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → zero ∈ R → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (∀x y ∈ R, x * y = y * x) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R, one * x = x) → (∀x ∈ R, x ≠ zero → ∃y ∈ R, x * y = one) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → explicit_Field

Axiom. (explicit_Field_E) We take the following as an axiom:

∀q : prop, (explicit_Field → (∀x y ∈ R, x + y ∈ R) → (∀x y z ∈ R, x + (y + z) = (x + y) + z) → (∀x y ∈ R, x + y = y + x) → (zero ∈ R) → (∀x ∈ R, zero + x = x) → (∀x ∈ R, ∃y ∈ R, x + y = zero) → (∀x y ∈ R, x * y ∈ R) → (∀x y z ∈ R, x * (y * z) = (x * y) * z) → (∀x y ∈ R, x * y = y * x) → (one ∈ R) → (one ≠ zero) → (∀x ∈ R, one * x = x) → (∀x ∈ R, x ≠ zero → ∃y ∈ R, x * y = one) → (∀x y z ∈ R, x * (y + z) = x * y + x * z) → q) → explicit_Field → q

Primitive. The name explicit_Field_minus is a term of type set → set.

Notation. We use - as a prefix operator with priority 358 corresponding to applying term explicit_Field_minus.

Axiom. (explicit_Field_minus_prop) We take the following as an axiom:

explicit_Field → ∀x ∈ R, - x ∈ R ∧ x + - x = zero

Axiom. (explicit_Field_minus_clos) We take the following as an axiom:

explicit_Field → ∀x ∈ R, - x ∈ R

Axiom. (explicit_Field_minus_R) We take the following as an axiom:

explicit_Field → ∀x ∈ R, x + - x = zero

Axiom. (explicit_Field_minus_L) We take the following as an axiom:

explicit_Field → ∀x ∈ R, - x + x = zero

Axiom. (explicit_Field_plus_cancelL) We take the following as an axiom:

explicit_Field → ∀x y z ∈ R, x + y = x + z → y = z

Axiom. (explicit_Field_plus_cancelR) We take the following as an axiom:

explicit_Field → ∀x y z ∈ R, x + z = y + z → x = y

Axiom. (explicit_Field_minus_invol) We take the following as an axiom:

explicit_Field → ∀x ∈ R, - - x = x

Axiom. (explicit_Field_minus_one_In) We take the following as an axiom:

explicit_Field → - one ∈ R

Axiom. (explicit_Field_zero_multR) We take the following as an axiom:

explicit_Field → ∀x ∈ R, x * zero = zero

Axiom. (explicit_Field_zero_multL) We take the following as an axiom:

explicit_Field → ∀x ∈ R, zero * x = zero

Axiom. (explicit_Field_minus_mult) We take the following as an axiom:

explicit_Field → ∀x ∈ R, - x = (- one) * x

Axiom. (explicit_Field_minus_one_square) We take the following as an axiom:

explicit_Field → (- one) * (- one) = one

Axiom. (explicit_Field_minus_square) We take the following as an axiom:

explicit_Field → ∀x ∈ R, (- x) * (- x) = x * x

Axiom. (explicit_Field_minus_zero) We take the following as an axiom:

explicit_Field → - zero = zero

Axiom. (explicit_Field_dist_R) We take the following as an axiom:

explicit_Field → ∀x y z ∈ R, (x + y) * z = x * z + y * z

Axiom. (explicit_Field_minus_plus_dist) We take the following as an axiom:

explicit_Field → ∀x y ∈ R, - (x + y) = - x + - y

Axiom. (explicit_Field_minus_mult_L) We take the following as an axiom:

explicit_Field → ∀x y ∈ R, (- x) * y = - (x * y)

Axiom. (explicit_Field_minus_mult_R) We take the following as an axiom:

explicit_Field → ∀x y ∈ R, x * (- y) = - (x * y)

Axiom. (explicit_Field_square_zero_inv) We take the following as an axiom:

explicit_Field → ∀x ∈ R, x * x = zero → x = zero

Axiom. (explicit_Field_mult_zero_inv) We take the following as an axiom:

explicit_Field → ∀x y ∈ R, x * y = zero → x = zero ∨ y = zero

Variable leq : set → set → prop

Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term leq.

Primitive. The name explicit_OrderedField is a term of type prop.

Axiom. (explicit_OrderedField_I) We take the following as an axiom:

explicit_Field → (∀x y z ∈ R, x ≤ y → y ≤ z → x ≤ z) → (∀x y ∈ R, x ≤ y ∧ y ≤ x ↔ x = y) → (∀x y ∈ R, x ≤ y ∨ y ≤ x) → (∀x y z ∈ R, x ≤ y → x + z ≤ y + z) → (∀x y ∈ R, zero ≤ x → zero ≤ y → zero ≤ x * y) → explicit_OrderedField

Axiom. (explicit_OrderedField_E) We take the following as an axiom:

∀q : prop, (explicit_OrderedField → explicit_Field → (∀x y z ∈ R, x ≤ y → y ≤ z → x ≤ z) → (∀x y ∈ R, x ≤ y ∧ y ≤ x ↔ x = y) → (∀x y ∈ R, x ≤ y ∨ y ≤ x) → (∀x y z ∈ R, x ≤ y → x + z ≤ y + z) → (∀x y ∈ R, zero ≤ x → zero ≤ y → zero ≤ x * y) → q) → explicit_OrderedField → q

Axiom. (explicit_OrderedField_minus_leq) We take the following as an axiom:

explicit_OrderedField → ∀x y ∈ R, x ≤ y → - y ≤ - x

Axiom. (explicit_OrderedField_square_nonneg) We take the following as an axiom:

explicit_OrderedField → ∀x ∈ R, zero ≤ x * x

Axiom. (explicit_OrderedField_sum_squares_nonneg) We take the following as an axiom:

explicit_OrderedField → ∀x y ∈ R, zero ≤ x * x + y * y

Axiom. (explicit_OrderedField_sum_nonneg_zero_inv) We take the following as an axiom:

explicit_OrderedField → ∀x y ∈ R, zero ≤ x → zero ≤ y → x + y = zero → x = zero ∧ y = zero

Axiom. (explicit_OrderedField_sum_squares_zero_inv) We take the following as an axiom:

explicit_OrderedField → ∀x y ∈ R, x * x + y * y = zero → x = zero ∧ y = zero

Axiom. (explicit_OrderedField_leq_refl) We take the following as an axiom:

explicit_OrderedField → ∀x ∈ R, x ≤ x

Axiom. (explicit_OrderedField_leq_antisym) We take the following as an axiom:

explicit_OrderedField → ∀x y ∈ R, x ≤ y → y ≤ x → x = y

Axiom. (explicit_OrderedField_leq_tra) We take the following as an axiom:

explicit_OrderedField → ∀x y z ∈ R, x ≤ y → y ≤ z → x ≤ z

Axiom. (explicit_OrderedField_leq_zero_one) We take the following as an axiom:

explicit_OrderedField → zero ≤ one

Definition. We define lt to be λx y ⇒ x ≤ y ∧ x ≠ y of type set → set → prop.

Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term lt.

Primitive. The name natOfOrderedField_p is a term of type set → prop.

Let N ≝ {n ∈ R|natOfOrderedField_p n}

Let Npos ≝ {n ∈ N|n ≠ zero}

Axiom. (explicit_Nats_natOfOrderedField) We take the following as an axiom:

explicit_OrderedField → explicit_Nats N zero (λm ⇒ m + one)

Axiom. (explicit_PosNats_natOfOrderedField) We take the following as an axiom:

explicit_OrderedField → explicit_Nats Npos one (λm ⇒ m + one)

Let Z ≝ {n ∈ R|- n ∈ Npos ∨ n = zero ∨ n ∈ Npos}

Definition. We define explicit_OrderedField_rationalp to be λx ⇒ ∃n ∈ Z, ∃m ∈ Npos, m * x = n of type set → prop.

Let Q ≝ {x ∈ R|explicit_OrderedField_rationalp x}

Axiom. (explicit_OrderedField_Npos_props) We take the following as an axiom:

explicit_OrderedField → ∀p : prop, (Npos ⊆ R → explicit_Nats Npos one (λm ⇒ m + one) → one ∈ Npos → (∀m ∈ Npos, m + one ≠ one) → (∀m ∈ Npos, ∀q : set → prop, q one → (∀n ∈ Npos, q (n + one)) → q m) → (∀n m ∈ Npos, explicit_Nats_one_plus Npos one (λm ⇒ m + one) n m = n + m) → (∀n m ∈ Npos, explicit_Nats_one_mult Npos one (λm ⇒ m + one) n m = n * m) → (∀n m ∈ Npos, n + m ∈ Npos) → (∀n m ∈ Npos, n * m ∈ Npos) → p) → p

Axiom. (explicit_OrderedField_Z_props) We take the following as an axiom:

explicit_OrderedField → ∀p : prop, ((∀n ∈ Npos, - n ∈ Z) → zero ∈ Z → Npos ⊆ Z → Z ⊆ R → (∀n ∈ Z, ∀q : prop, (- n ∈ Npos → q) → (n = zero → q) → (n ∈ Npos → q) → q) → one ∈ Z → - one ∈ Z → (∀m ∈ Z, - m ∈ Z) → (∀n m ∈ Z, n + m ∈ Z) → (∀n m ∈ Z, n * m ∈ Z) → p) → p

Axiom. (explicit_OrderedField_Q_props) We take the following as an axiom:

explicit_OrderedField → ∀p : prop, (Q ⊆ R → (∀x ∈ Q, ∀q : prop, (x ∈ R → ∀n ∈ Z, ∀m ∈ Npos, m * x = n → q) → q) → (∀x ∈ R, ∀n ∈ Z, ∀m ∈ Npos, m * x = n → x ∈ Q) → p) → p

Primitive. The name explicit_Reals is a term of type prop.

Axiom. (explicit_Reals_I) We take the following as an axiom:

explicit_OrderedField → (∀x y ∈ R, zero < x → zero ≤ y → ∃n ∈ N, y ≤ n * x) → (∀a b ∈ R^N, (∀n ∈ N, a n ≤ b n ∧ a n ≤ a (n + one) ∧ b (n + one) ≤ b n) → ∃x ∈ R, ∀n ∈ N, a n ≤ x ∧ x ≤ b n) → explicit_Reals

Axiom. (explicit_Reals_E) We take the following as an axiom:

∀q : prop, (explicit_Reals → explicit_OrderedField → (∀x y ∈ R, zero < x → zero ≤ y → ∃n ∈ N, y ≤ n * x) → (∀a b ∈ R^N, (∀n ∈ N, a n ≤ b n ∧ a n ≤ a (n + one) ∧ b (n + one) ≤ b n) → ∃x ∈ R, ∀n ∈ N, a n ≤ x ∧ x ≤ b n) → q) → explicit_Reals → q

Axiom. (explicit_Reals_characteristic_0) We take the following as an axiom:

explicit_Reals → ∀n ∈ ω, nat_primrec one (λ_ r ⇒ plus one r) n ≠ zero

End of Section explicit_Reals

Definition. We define CRing_with_id_carrier to be λRs ⇒ Rs 0 of type set → set.

Definition. We define CRing_with_id_plus to be λRs ⇒ decode_b (Rs 1) of type set → set → set → set.

Definition. We define CRing_with_id_mult to be λRs ⇒ decode_b (Rs 2) of type set → set → set → set.

Definition. We define CRing_with_id_zero to be λRs ⇒ Rs 3 of type set → set.

Definition. We define CRing_with_id_one to be λRs ⇒ Rs 4 of type set → set.

Beginning of Section CRing_with_id

Variable Rs : set

Hypothesis HRs : CRing_with_id Rs

Let R : set ≝ CRing_with_id_carrier Rs

Let zero : set ≝ CRing_with_id_zero Rs

Let one : set ≝ CRing_with_id_one Rs

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term CRing_with_id_plus Rs.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term CRing_with_id_mult Rs.

Axiom. (CRing_with_id_eta) We take the following as an axiom:

Rs = pack_b_b_e_e R (CRing_with_id_plus Rs) (CRing_with_id_mult Rs) zero one

Axiom. (CRing_with_id_explicit_CRing_with_id) We take the following as an axiom:

explicit_CRing_with_id R zero one (CRing_with_id_plus Rs) (CRing_with_id_mult Rs)

Axiom. (CRing_with_id_zero_In) We take the following as an axiom:

zero ∈ R

Axiom. (CRing_with_id_one_In) We take the following as an axiom:

one ∈ R

Axiom. (CRing_with_id_plus_clos) We take the following as an axiom:

∀x y ∈ R, x + y ∈ R

Axiom. (CRing_with_id_mult_clos) We take the following as an axiom:

∀x y ∈ R, x * y ∈ R

Axiom. (CRing_with_id_plus_assoc) We take the following as an axiom:

∀x y z ∈ R, x + (y + z) = (x + y) + z

Axiom. (CRing_with_id_plus_com) We take the following as an axiom:

∀x y ∈ R, x + y = y + x

Axiom. (CRing_with_id_zero_L) We take the following as an axiom:

∀x ∈ R, zero + x = x

Axiom. (CRing_with_id_plus_inv) We take the following as an axiom:

∀x ∈ R, ∃y ∈ R, x + y = zero

Axiom. (CRing_with_id_mult_assoc) We take the following as an axiom:

∀x y z ∈ R, x * (y * z) = (x * y) * z

Axiom. (CRing_with_id_mult_com) We take the following as an axiom:

∀x y ∈ R, x * y = y * x

Axiom. (CRing_with_id_one_neq_zero) We take the following as an axiom:

one ≠ zero

Axiom. (CRing_with_id_one_L) We take the following as an axiom:

∀x ∈ R, one * x = x

Axiom. (CRing_with_id_distr_L) We take the following as an axiom:

∀x y z ∈ R, x * (y + z) = x * y + x * z

Definition. We define CRing_with_id_omega_exp to be λx ⇒ nat_primrec one (λk r ⇒ x * r) of type set → set → set.

Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term CRing_with_id_omega_exp.

Axiom. (CRing_with_id_omega_exp_0) We take the following as an axiom:

∀x, x ^ 0 = one

Axiom. (CRing_with_id_omega_exp_S) We take the following as an axiom:

∀x, ∀n ∈ ω, x ^ (ordsucc n) = x * x ^ n

Axiom. (CRing_with_id_omega_exp_1) We take the following as an axiom:

∀x ∈ R, x ^ 1 = x

Axiom. (CRing_with_id_omega_exp_clos) We take the following as an axiom:

∀x ∈ R, ∀n ∈ ω, x ^ n ∈ R

Definition. We define CRing_with_id_eval_poly to be λn cs x ⇒ nat_primrec zero (λm r ⇒ cs m * x ^ m + r) n of type set → set → set → set.

Axiom. (CRing_with_id_eval_poly_clos) We take the following as an axiom:

∀n ∈ ω, ∀cs ∈ Rⁿ, ∀x ∈ R, CRing_with_id_eval_poly n cs x ∈ R

End of Section CRing_with_id

Beginning of Section explicit_Reals

Variable R : set

Variable zero one : set

Variable plus mult : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Notation. We use - as a prefix operator with priority 358 corresponding to applying term explicit_Field_minus R zero one plus mult.

Variable leq : set → set → prop

Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term leq.

Let N ≝ {n ∈ R|natOfOrderedField_p R zero one plus mult leq n}

Let Npos ≝ {n ∈ N|n ≠ zero}

Let Z ≝ {n ∈ R|- n ∈ Npos ∨ n = zero ∨ n ∈ Npos}

Let Q ≝ {x ∈ R|explicit_OrderedField_rationalp R zero one plus mult leq x}

Axiom. (explicit_OrderedField_explicit_Field_Q) We take the following as an axiom:

explicit_OrderedField R zero one plus mult leq → explicit_Field Q zero one plus mult

Axiom. (explicit_OrderedField_sub) We take the following as an axiom:

explicit_OrderedField R zero one plus mult leq → ∀R' ⊆ R, zero ∈ R' → one ∈ R' → (∀x y ∈ R', x + y ∈ R') → (∀x ∈ R', - x ∈ R') → (∀x y ∈ R', x * y ∈ R') → (∀x ∈ R', x ≠ zero → ∃y ∈ R', x * y = one) → explicit_OrderedField R' zero one plus mult leq

Axiom. (explicit_Reals_sub) We take the following as an axiom:

Beginning of Section explicit_Reals_Q_min_props

Variable R' : set

Let N' ≝ {n ∈ R'|natOfOrderedField_p R' zero one plus mult leq n}

Let Npos' ≝ {n ∈ N'|n ≠ zero}

Let Z' ≝ {n ∈ R'|explicit_Field_minus R' zero one plus mult n ∈ Npos' ∨ n = zero ∨ n ∈ Npos'}

Let Q' ≝ {x ∈ R'|explicit_OrderedField_rationalp R' zero one plus mult leq x}

Axiom. (explicit_Reals_Q_min_props) We take the following as an axiom:

explicit_OrderedField R zero one plus mult leq → R' ⊆ R → explicit_Field R' zero one plus mult → ∀p : prop, ((∀x ∈ R', explicit_Field_minus R' zero one plus mult x = - x) → (∀x ∈ R', - x ∈ R') → N = N' → Npos = Npos' → Z = Z' → Q = Q' → p) → p

End of Section explicit_Reals_Q_min_props

Axiom. (explicit_Reals_Q_min) We take the following as an axiom:

explicit_OrderedField R zero one plus mult leq → ∀R' ⊆ R, explicit_Field R' zero one plus mult → Q ⊆ R'

End of Section explicit_Reals

Beginning of Section explicit_Field_transfer

Variable R : set

Variable zero one : set

Variable plus mult : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Variable R' : set

Variable zero' one' : set

Variable plus' mult' : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus'.

Notation. We use ⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term mult'.

Variable f : set → set

Axiom. (explicit_Field_transfer) We take the following as an axiom:

explicit_Field R zero one plus mult → bij R R' f → f zero = zero' → f one = one' → (∀x y ∈ R, f (x + y) = f x + f y) → (∀x y ∈ R, f (x * y) = f x ⨯ f y) → explicit_Field R' zero' one' plus' mult'

End of Section explicit_Field_transfer

Beginning of Section explicit_Field_RepIndep2

Variable R : set

Variable zero one : set

Variable plus mult : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Variable plus' mult' : set → set → set

Notation. We use + as an infix operator with priority 355 and which associates to the right corresponding to applying term plus'.

Notation. We use ⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term mult'.

Hypothesis Hpp' : ∀a b ∈ R, a + b = a + b

Hypothesis Hmm' : ∀a b ∈ R, a * b = a ⨯ b

Axiom. (explicit_Field_repindep) We take the following as an axiom:

explicit_Field R zero one plus mult ↔ explicit_Field R zero one plus' mult'

End of Section explicit_Field_RepIndep2

Definition. We define Field to be λF ⇒ struct_b_b_e_e F ∧ unpack_b_b_e_e_o F (λQ plus mult zero one ⇒ explicit_Field Q zero one plus mult) of type set → prop.

Axiom. (Field_unpack_eq) We take the following as an axiom:

∀R, ∀plus mult : set → set → set, ∀zero one, unpack_b_b_e_e_o (pack_b_b_e_e R plus mult zero one) (λR plus mult zero one ⇒ explicit_Field R zero one plus mult) = explicit_Field R zero one plus mult

Axiom. (Field_is_CRing_with_id) We take the following as an axiom:

∀F, Field F → CRing_with_id F

Beginning of Section explicit_OrderedField_transfer

Variable R : set

Variable zero one : set

Variable plus mult : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Variable leq : set → set → prop

Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term leq.

Variable R' : set

Variable zero' one' : set

Variable plus' mult' : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus'.

Notation. We use ⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term mult'.

Variable leq' : set → set → prop

Variable f : set → set

Axiom. (explicit_OrderedField_transfer) We take the following as an axiom:

explicit_OrderedField R zero one plus mult leq → bij R R' f → f zero = zero' → f one = one' → (∀x y ∈ R, f (x + y) = f x + f y) → (∀x y ∈ R, f (x * y) = f x ⨯ f y) → (∀x y ∈ R, x ≤ y ↔ leq' (f x) (f y)) → explicit_OrderedField R' zero' one' plus' mult' leq'

End of Section explicit_OrderedField_transfer

Definition. We define Field_carrier to be λFs ⇒ Fs 0 of type set → set.

Definition. We define Field_plus to be λFs ⇒ decode_b (Fs 1) of type set → set → set → set.

Definition. We define Field_mult to be λFs ⇒ decode_b (Fs 2) of type set → set → set → set.

Definition. We define Field_zero to be λFs ⇒ Fs 3 of type set → set.

Definition. We define Field_one to be λFs ⇒ Fs 4 of type set → set.

Primitive. The name Field_minus is a term of type set → set → set.

Beginning of Section Field

Variable Fs : set

Hypothesis HFs : Field Fs

Let F : set ≝ Field_carrier Fs

Let zero : set ≝ Field_zero Fs

Let one : set ≝ Field_one Fs

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term Field_plus Fs.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term Field_mult Fs.

Notation. We use - as a prefix operator with priority 358 corresponding to applying term Field_minus Fs.

Axiom. (Field_eta) We take the following as an axiom:

Fs = pack_b_b_e_e F (Field_plus Fs) (Field_mult Fs) zero one

Axiom. (Field_explicit_Field) We take the following as an axiom:

explicit_Field F zero one (Field_plus Fs) (Field_mult Fs)

Axiom. (Field_zero_In) We take the following as an axiom:

zero ∈ F

Axiom. (Field_one_In) We take the following as an axiom:

one ∈ F

Axiom. (Field_plus_clos) We take the following as an axiom:

∀x y ∈ F, x + y ∈ F

Axiom. (Field_mult_clos) We take the following as an axiom:

∀x y ∈ F, x * y ∈ F

Axiom. (Field_plus_assoc) We take the following as an axiom:

∀x y z ∈ F, x + (y + z) = (x + y) + z

Axiom. (Field_plus_com) We take the following as an axiom:

∀x y ∈ F, x + y = y + x

Axiom. (Field_zero_L) We take the following as an axiom:

∀x ∈ F, zero + x = x

Axiom. (Field_plus_inv) We take the following as an axiom:

∀x ∈ F, ∃y ∈ F, x + y = zero

Axiom. (Field_mult_assoc) We take the following as an axiom:

∀x y z ∈ F, x * (y * z) = (x * y) * z

Axiom. (Field_mult_com) We take the following as an axiom:

∀x y ∈ F, x * y = y * x

Axiom. (Field_one_neq_zero) We take the following as an axiom:

one ≠ zero

Axiom. (Field_one_L) We take the following as an axiom:

∀x ∈ F, one * x = x

Axiom. (Field_mult_inv_L) We take the following as an axiom:

∀x ∈ F, x ≠ zero → ∃y ∈ F, x * y = one

Axiom. (Field_distr_L) We take the following as an axiom:

∀x y z ∈ F, x * (y + z) = x * y + x * z

Primitive. The name Field_div is a term of type set → set → set.

Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term Field_div.

Axiom. (Field_div_prop) We take the following as an axiom:

∀x ∈ F, ∀y ∈ F ∖ {zero}, x :/: y ∈ F ∧ x = y * (x :/: y)

Axiom. (Field_div_clos) We take the following as an axiom:

∀x ∈ F, ∀y ∈ F ∖ {zero}, x :/: y ∈ F

Axiom. (Field_mult_div) We take the following as an axiom:

∀x ∈ F, ∀y ∈ F ∖ {zero}, x = y * (x :/: y)

Axiom. (Field_div_undef1) We take the following as an axiom:

∀x y, x ∉ F → x :/: y = 0

Axiom. (Field_div_undef2) We take the following as an axiom:

∀x y, y ∉ F → x :/: y = 0

Axiom. (Field_div_undef3) We take the following as an axiom:

∀x, x :/: zero = 0

Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term CRing_with_id_omega_exp Fs.

Axiom. (Field_omega_exp_0) We take the following as an axiom:

∀x, x ^ 0 = one

Axiom. (Field_omega_exp_S) We take the following as an axiom:

∀x, ∀n ∈ ω, x ^ (ordsucc n) = x * x ^ n

Axiom. (Field_omega_exp_1) We take the following as an axiom:

∀x ∈ F, x ^ 1 = x

Axiom. (Field_omega_exp_clos) We take the following as an axiom:

∀x ∈ F, ∀n ∈ ω, x ^ n ∈ F

Axiom. (Field_eval_poly_clos) We take the following as an axiom:

∀n ∈ ω, ∀cs ∈ Fⁿ, ∀x ∈ F, CRing_with_id_eval_poly Fs n cs x ∈ F

Axiom. (Field_plus_cancelL) We take the following as an axiom:

∀x y z ∈ F, x + y = x + z → y = z

Axiom. (Field_plus_cancelR) We take the following as an axiom:

∀x y z ∈ F, x + z = y + z → x = y

Axiom. (Field_minus_eq) We take the following as an axiom:

∀x ∈ F, - x = explicit_Field_minus F zero one (Field_plus Fs) (Field_mult Fs) x

Axiom. (Field_minus_undef) We take the following as an axiom:

∀x, x ∉ F → - x = 0

Axiom. (Field_minus_clos) We take the following as an axiom:

∀x ∈ F, - x ∈ F

Axiom. (Field_minus_R) We take the following as an axiom:

∀x ∈ F, x + - x = zero

Axiom. (Field_minus_L) We take the following as an axiom:

∀x ∈ F, - x + x = zero

Axiom. (Field_minus_invol) We take the following as an axiom:

∀x ∈ F, - - x = x

Axiom. (Field_minus_one_In) We take the following as an axiom:

- one ∈ F

Axiom. (Field_zero_multR) We take the following as an axiom:

∀x ∈ F, x * zero = zero

Axiom. (Field_zero_multL) We take the following as an axiom:

∀x ∈ F, zero * x = zero

Axiom. (Field_minus_mult) We take the following as an axiom:

∀x ∈ F, - x = (- one) * x

Axiom. (Field_minus_one_square) We take the following as an axiom:

(- one) * (- one) = one

Axiom. (Field_minus_square) We take the following as an axiom:

∀x ∈ F, (- x) * (- x) = x * x

Axiom. (Field_minus_zero) We take the following as an axiom:

- zero = zero

Axiom. (Field_dist_R) We take the following as an axiom:

∀x y z ∈ F, (x + y) * z = x * z + y * z

Axiom. (Field_minus_plus_dist) We take the following as an axiom:

∀x y ∈ F, - (x + y) = - x + - y

Axiom. (Field_minus_mult_L) We take the following as an axiom:

∀x y ∈ F, (- x) * y = - (x * y)

Axiom. (Field_minus_mult_R) We take the following as an axiom:

∀x y ∈ F, x * (- y) = - (x * y)

Axiom. (Field_square_zero_inv) We take the following as an axiom:

∀x ∈ F, x * x = zero → x = zero

Axiom. (Field_mult_zero_inv) We take the following as an axiom:

∀x y ∈ F, x * y = zero → x = zero ∨ y = zero

End of Section Field

Beginning of Section Field2

Variable Fs : set

Variable Fs' : set

Let F : set ≝ Field_carrier Fs

Let zero : set ≝ Field_zero Fs

Let one : set ≝ Field_one Fs

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term Field_plus Fs.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term Field_mult Fs.

Let F' : set ≝ Field_carrier Fs'

Let zero' : set ≝ Field_zero Fs'

Let one' : set ≝ Field_one Fs'

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term Field_plus Fs'.

Notation. We use ⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term Field_mult Fs'.

Notation. We use - as a prefix operator with priority 358 corresponding to applying term Field_minus Fs.

Notation. We use :-: as a prefix operator with priority 358 corresponding to applying term Field_minus Fs'.

Primitive. The name subfield is a term of type prop.

Axiom. (subfield_I) We take the following as an axiom:

Field Fs → Field Fs' → F ⊆ F' → zero = zero' → one = one' → (∀a b ∈ F, a + b = a + b) → (∀a b ∈ F, a * b = a ⨯ b) → subfield

Axiom. (subfield_E) We take the following as an axiom:

subfield → ∀p : prop, (Field Fs → Field Fs' → F ⊆ F' → zero = zero' → one = one' → (∀a b ∈ F, a + b = a + b) → (∀a b ∈ F, a * b = a ⨯ b) → p) → p

Primitive. The name Field_Hom is a term of type set → prop.

Axiom. (Field_Hom_I) We take the following as an axiom:

∀g, Field Fs → Field Fs' → g ∈ F'^F → g zero = zero' → g one = one' → (∀a b ∈ F, g (a + b) = g a + g b) → (∀a b ∈ F, g (a * b) = g a ⨯ g b) → Field_Hom g

Axiom. (Field_Hom_E) We take the following as an axiom:

∀g, Field_Hom g → ∀p : prop, (Field Fs → Field Fs' → g ∈ F'^F → g zero = zero' → g one = one' → (∀a b ∈ F, g (a + b) = g a + g b) → (∀a b ∈ F, g (a * b) = g a ⨯ g b) → (∀a ∈ F, g (- a) = :-: g a) → (∀a ∈ F, g a = zero' → a = zero) → (∀a b ∈ F, g a = g b → a = b) → (∀a ∈ F, ∀n ∈ ω, g (CRing_with_id_omega_exp Fs a n) = CRing_with_id_omega_exp Fs' (g a) n) → p) → p

Axiom. (Field_Hom_inj) We take the following as an axiom:

∀g, Field_Hom g → ∀a b ∈ F, g a = g b → a = b

End of Section Field2

Axiom. (subfield_refl) We take the following as an axiom:

∀Fs, Field Fs → subfield Fs Fs

Axiom. (subfield_tra) We take the following as an axiom:

∀Fs Fs' Fs'', subfield Fs Fs' → subfield Fs' Fs'' → subfield Fs Fs''

Primitive. The name Field_extension_by_1 is a term of type set → set → set → prop.

Axiom. (Field_extension_by_1_I) We take the following as an axiom:

∀Fs Fs' a, subfield Fs Fs' → a ∈ Field_carrier Fs' ∖ Field_carrier Fs → (∀Fs'', subfield Fs Fs'' → a ∈ Field_carrier Fs'' → subfield Fs' Fs'') → Field_extension_by_1 Fs Fs' a

Axiom. (Field_extension_by_1_E) We take the following as an axiom:

∀Fs Fs' a, Field_extension_by_1 Fs Fs' a → ∀p : prop, (subfield Fs Fs' → a ∈ Field_carrier Fs' ∖ Field_carrier Fs → (∀Fs'', subfield Fs Fs'' → a ∈ Field_carrier Fs'' → subfield Fs' Fs'') → p) → p

Primitive. The name radical_field_extension is a term of type set → set → prop.

Axiom. (radical_field_extension_I) We take the following as an axiom:

∀Fs Fs', ∀r ∈ ω, ∀Fseq, Fseq 0 = Fs → Fseq r = Fs' → (∀i ∈ ordsucc r, Field (Fseq i)) → (∀i ∈ r, ∃a ∈ Field_carrier (Fseq (ordsucc i)), ∃n ∈ ω, CRing_with_id_omega_exp (Fseq (ordsucc i)) a n ∈ Field_carrier (Fseq i) ∧ Field_extension_by_1 (Fseq i) (Fseq (ordsucc i)) a) → radical_field_extension Fs Fs'

Axiom. (radical_field_extension_E) We take the following as an axiom:

∀Fs Fs', radical_field_extension Fs Fs' → ∀p : prop, (Field Fs → Field Fs' → subfield Fs Fs' → ∀r ∈ ω, ∀Fseq, Fseq 0 = Fs → Fseq r = Fs' → (∀i ∈ ordsucc r, Field (Fseq i)) → (∀i ∈ ordsucc r, ∀j ∈ ordsucc i, subfield (Fseq j) (Fseq i)) → (∀i ∈ r, ∃a ∈ Field_carrier (Fseq (ordsucc i)), ∃n ∈ ω, CRing_with_id_omega_exp (Fseq (ordsucc i)) a n ∈ Field_carrier (Fseq i) ∧ Field_extension_by_1 (Fseq i) (Fseq (ordsucc i)) a) → p) → p

Primitive. The name Field_automorphism_fixing is a term of type set → set → set → prop.

Axiom. (Field_automorphism_fixing_I) We take the following as an axiom:

∀K F f, subfield F K → Field_Hom K K f → (∀y ∈ K 0, ∃x ∈ K 0, f x = y) → (∀x ∈ F 0, f x = x) → Field_automorphism_fixing K F f

Axiom. (Field_automorphism_fixing_E) We take the following as an axiom:

∀K F f, Field_automorphism_fixing K F f → ∀p : prop, (subfield F K → Field_Hom K K f → (∀y ∈ K 0, ∃x ∈ K 0, f x = y) → (∀x ∈ F 0, f x = x) → p) → p

Definition. We define lam_comp to be λA f g ⇒ λx ∈ A ⇒ f (g x) of type set → set → set → set.

Definition. We define lam_id to be λA ⇒ λx ∈ A ⇒ x of type set → set.

Axiom. (lam_comp_exp_In) We take the following as an axiom:

∀A B C, ∀f ∈ B^A, ∀g ∈ C^B, lam_comp A g f ∈ C^A

Axiom. (lam_id_exp_In) We take the following as an axiom:

∀A, lam_id A ∈ A^A

Axiom. (lam_comp_assoc) We take the following as an axiom:

∀A B, ∀f ∈ B^A, ∀g h, lam_comp A h (lam_comp A g f) = lam_comp A (lam_comp B h g) f

Axiom. (lam_comp_id_L) We take the following as an axiom:

∀A B, ∀f ∈ B^A, lam_comp A (lam_id B) f = f

Axiom. (lam_comp_id_R) We take the following as an axiom:

∀A B, ∀f ∈ B^A, lam_comp A f (lam_id A) = f

Axiom. (Field_Hom_id) We take the following as an axiom:

∀F, Field F → Field_Hom F F (lam_id (F 0))

Axiom. (Field_Hom_comp) We take the following as an axiom:

∀F F' F'' g h, Field_Hom F F' g → Field_Hom F' F'' h → Field_Hom F F'' (lam_comp (F 0) h g)

Definition. We define Galois_Group to be λK F ⇒ pack_b {f ∈ K 0^{K 0}|Field_automorphism_fixing K F f} (lam_comp (K 0)) of type set → set → set.

Axiom. (Galois_Group_0) We take the following as an axiom:

∀K F, Galois_Group K F 0 = {f ∈ K 0^{K 0}|Field_automorphism_fixing K F f}

Axiom. (Galois_Group_Group) We take the following as an axiom:

∀F K, subfield F K → Group (Galois_Group K F)

Beginning of Section explicit_Reals_transfer

Variable R : set

Variable zero one : set

Variable plus mult : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Variable leq : set → set → prop

Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term leq.

Variable R' : set

Variable zero' one' : set

Variable plus' mult' : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus'.

Notation. We use ⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term mult'.

Variable leq' : set → set → prop

Variable f : set → set

Axiom. (explicit_Reals_transfer) We take the following as an axiom:

explicit_Reals R zero one plus mult leq → bij R R' f → f zero = zero' → f one = one' → (∀x y ∈ R, f (x + y) = f x + f y) → (∀x y ∈ R, f (x * y) = f x ⨯ f y) → (∀x y ∈ R, x ≤ y ↔ leq' (f x) (f y)) → explicit_Reals R' zero' one' plus' mult' leq'

End of Section explicit_Reals_transfer

Beginning of Section explicit_Complex

Variable C : set

Variable Re Im : set → set

Variable zero one i : set

Variable plus mult : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Let R ≝ {z ∈ C|Re z = z}

Primitive. The name explicit_Complex is a term of type prop.

Axiom. (explicit_Complex_I) We take the following as an axiom:

explicit_Field C zero one plus mult → (∃leq : set → set → prop, explicit_Reals R zero one plus mult leq) → (∀z ∈ C, Im z ∈ R) → (i ∈ C) → (∀z ∈ C, Re z ∈ C) → (∀z ∈ C, Im z ∈ C) → (∀z ∈ C, z = Re z + i * Im z) → (∀z w ∈ C, Re z = Re w → Im z = Im w → z = w) → (i * i + one = zero) → explicit_Complex

End of Section explicit_Complex

Beginning of Section RealsToComplex

Variable R : set

Variable zero one : set

Variable plus mult : set → set → set

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Notation. We use - as a prefix operator with priority 358 corresponding to applying term explicit_Field_minus R zero one plus mult.

Variable leq : set → set → prop

Variable pa : set → set → set

Let C : set ≝ ReplSep2 R (λ_ ⇒ R) (λx y ⇒ True) pa

Let Re : set → set ≝ λz ⇒ Eps_i (λx ⇒ x ∈ R ∧ ∃y ∈ R, z = pa x y)

Let Im : set → set ≝ λz ⇒ Eps_i (λy ⇒ y ∈ R ∧ z = pa (Re z) y)

Let Re' : set → set ≝ λz ⇒ pa (Re z) zero

Let Im' : set → set ≝ λz ⇒ pa (Im z) zero

Let R' ≝ {z ∈ C|Re' z = z}

Let zero' : set ≝ pa zero zero

Let one' : set ≝ pa one zero

Let i' : set ≝ pa zero one

Let plus' : set → set → set ≝ λz w ⇒ pa (Re z + Re w) (Im z + Im w)

Let mult' : set → set → set ≝ λz w ⇒ pa (Re z * Re w + - (Im z * Im w)) (Re z * Im w + Im z * Re w)

Axiom. (explicit_RealsToComplex) We take the following as an axiom:

explicit_Reals R zero one plus mult leq → (∀x1 y1 x2 y2 ∈ R, pa x1 y1 = pa x2 y2 → x1 = x2 ∧ y1 = y2) → explicit_Complex C Re' Im' zero' one' i' plus' mult'

Axiom. (explicit_RealsToComplex_exact_Subq) We take the following as an axiom:

explicit_Reals R zero one plus mult leq → (∀x1 y1 x2 y2 ∈ R, pa x1 y1 = pa x2 y2 → x1 = x2 ∧ y1 = y2) → (∀x ∈ R, pa x zero = x) → explicit_Complex C Re' Im' zero' one' i' plus' mult' ∧ R ⊆ C ∧ (∀x ∈ R, Re x = x) ∧ zero' = zero ∧ one' = one ∧ (∀x y ∈ R, plus' x y = x + y) ∧ (∀x y ∈ R, mult' x y = x * y)

End of Section RealsToComplex

Beginning of Section SurrealArithmetic

Primitive. The name minus_SNo is a term of type set → set.

Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.

Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.

Axiom. (minus_SNo_eq) We take the following as an axiom:

∀x, SNo x → - x = SNoCut {- z|z ∈ SNoR x} {- w|w ∈ SNoL x}

Axiom. (minus_SNo_prop1) We take the following as an axiom:

∀x, SNo x → SNo (- x) ∧ (∀u ∈ SNoL x, - x < - u) ∧ (∀u ∈ SNoR x, - u < - x) ∧ SNoCutP {- z|z ∈ SNoR x} {- w|w ∈ SNoL x}

Axiom. (SNo_minus_SNo) We take the following as an axiom:

∀x, SNo x → SNo (- x)

Axiom. (minus_SNo_Lt_contra) We take the following as an axiom:

∀x y, SNo x → SNo y → x < y → - y < - x

Axiom. (minus_SNo_Le_contra) We take the following as an axiom:

∀x y, SNo x → SNo y → x ≤ y → - y ≤ - x

Axiom. (minus_SNo_SNoCutP) We take the following as an axiom:

∀x, SNo x → SNoCutP {- z|z ∈ SNoR x} {- w|w ∈ SNoL x}

Axiom. (minus_SNo_SNoCutP_gen) We take the following as an axiom:

∀L R, SNoCutP L R → SNoCutP {- z|z ∈ R} {- w|w ∈ L}

Axiom. (minus_SNo_Lev_lem1) We take the following as an axiom:

∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, SNoLev (- x) ⊆ SNoLev x

Axiom. (minus_SNo_Lev_lem2) We take the following as an axiom:

∀x, SNo x → SNoLev (- x) ⊆ SNoLev x

Axiom. (minus_SNo_invol) We take the following as an axiom:

∀x, SNo x → - - x = x

Axiom. (minus_SNo_Lev) We take the following as an axiom:

∀x, SNo x → SNoLev (- x) = SNoLev x

Axiom. (minus_SNo_SNo_) We take the following as an axiom:

∀alpha, ordinal alpha → ∀x, SNo_ alpha x → SNo_ alpha (- x)

Axiom. (minus_SNo_SNoS_) We take the following as an axiom:

∀alpha, ordinal alpha → ∀x, x ∈ SNoS_ alpha → - x ∈ SNoS_ alpha

Axiom. (minus_SNoCut_eq_lem) We take the following as an axiom:

∀v, SNo v → ∀L R, SNoCutP L R → v = SNoCut L R → - v = SNoCut {- z|z ∈ R} {- w|w ∈ L}

Axiom. (minus_SNoCut_eq) We take the following as an axiom:

∀L R, SNoCutP L R → - SNoCut L R = SNoCut {- z|z ∈ R} {- w|w ∈ L}

Axiom. (minus_SNo_Lt_contra1) We take the following as an axiom:

∀x y, SNo x → SNo y → - x < y → - y < x

Axiom. (minus_SNo_Lt_contra2) We take the following as an axiom:

∀x y, SNo x → SNo y → x < - y → y < - x

Axiom. (minus_SNo_Lt_contra3) We take the following as an axiom:

∀x y, SNo x → SNo y → - x < - y → y < x

Axiom. (minus_SNo_0) We take the following as an axiom:

- 0 = 0

Axiom. (SNo_momega) We take the following as an axiom:

SNo (- ω)

Axiom. (mordinal_SNo) We take the following as an axiom:

∀alpha, ordinal alpha → SNo (- alpha)

Axiom. (mordinal_SNoLev) We take the following as an axiom:

∀alpha, ordinal alpha → SNoLev (- alpha) = alpha

Axiom. (mordinal_SNoLev_min) We take the following as an axiom:

∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ alpha → - alpha < z

Axiom. (mordinal_SNoLev_min_2) We take the following as an axiom:

∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ ordsucc alpha → - alpha ≤ z

Axiom. (minus_SNo_SNoS_omega) We take the following as an axiom:

∀x ∈ SNoS_ ω, - x ∈ SNoS_ ω

Primitive. The name add_SNo is a term of type set → set → set.

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.

Axiom. (add_SNo_eq) We take the following as an axiom:

∀x, SNo x → ∀y, SNo y → x + y = SNoCut ({w + y|w ∈ SNoL x} ∪ {x + w|w ∈ SNoL y}) ({z + y|z ∈ SNoR x} ∪ {x + z|z ∈ SNoR y})

Axiom. (add_SNo_prop1) We take the following as an axiom:

∀x y, SNo x → SNo y → SNo (x + y) ∧ (∀u ∈ SNoL x, u + y < x + y) ∧ (∀u ∈ SNoR x, x + y < u + y) ∧ (∀u ∈ SNoL y, x + u < x + y) ∧ (∀u ∈ SNoR y, x + y < x + u) ∧ SNoCutP ({w + y|w ∈ SNoL x} ∪ {x + w|w ∈ SNoL y}) ({z + y|z ∈ SNoR x} ∪ {x + z|z ∈ SNoR y})

Axiom. (SNo_add_SNo) We take the following as an axiom:

∀x y, SNo x → SNo y → SNo (x + y)

Axiom. (SNo_add_SNo_3) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → SNo (x + y + z)

Axiom. (SNo_add_SNo_4) We take the following as an axiom:

∀x y z w, SNo x → SNo y → SNo z → SNo w → SNo (x + y + z + w)

Axiom. (add_SNo_Lt1) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x < z → x + y < z + y

Axiom. (add_SNo_Le1) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x ≤ z → x + y ≤ z + y

Axiom. (add_SNo_Lt2) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → y < z → x + y < x + z

Axiom. (add_SNo_Le2) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → y ≤ z → x + y ≤ x + z

Axiom. (add_SNo_Lt3a) We take the following as an axiom:

∀x y z w, SNo x → SNo y → SNo z → SNo w → x < z → y ≤ w → x + y < z + w

Axiom. (add_SNo_Lt3b) We take the following as an axiom:

∀x y z w, SNo x → SNo y → SNo z → SNo w → x ≤ z → y < w → x + y < z + w

Axiom. (add_SNo_Lt3) We take the following as an axiom:

∀x y z w, SNo x → SNo y → SNo z → SNo w → x < z → y < w → x + y < z + w

Axiom. (add_SNo_Le3) We take the following as an axiom:

∀x y z w, SNo x → SNo y → SNo z → SNo w → x ≤ z → y ≤ w → x + y ≤ z + w

Axiom. (add_SNo_SNoCutP) We take the following as an axiom:

∀x y, SNo x → SNo y → SNoCutP ({w + y|w ∈ SNoL x} ∪ {x + w|w ∈ SNoL y}) ({z + y|z ∈ SNoR x} ∪ {x + z|z ∈ SNoR y})

Axiom. (add_SNo_SNoCutP_gen) We take the following as an axiom:

∀Lx Rx Ly Ry, SNoCutP Lx Rx → SNoCutP Ly Ry → SNoCutP ({w + SNoCut Ly Ry|w ∈ Lx} ∪ {SNoCut Lx Rx + w|w ∈ Ly}) ({z + SNoCut Ly Ry|z ∈ Rx} ∪ {SNoCut Lx Rx + z|z ∈ Ry})

Axiom. (add_SNo_com) We take the following as an axiom:

∀x y, SNo x → SNo y → x + y = y + x

Axiom. (add_SNo_0L) We take the following as an axiom:

∀x, SNo x → 0 + x = x

Axiom. (add_SNo_0R) We take the following as an axiom:

∀x, SNo x → x + 0 = x

Axiom. (add_SNo_minus_SNo_linv) We take the following as an axiom:

∀x, SNo x → - x + x = 0

Axiom. (SNo_add_SNo_3c) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → SNo (x + y + - z)

Axiom. (add_SNo_minus_SNo_rinv) We take the following as an axiom:

∀x, SNo x → x + - x = 0

Axiom. (add_SNo_ordinal_SNoCutP) We take the following as an axiom:

∀alpha, ordinal alpha → ∀beta, ordinal beta → SNoCutP ({x + beta|x ∈ SNoS_ alpha} ∪ {alpha + x|x ∈ SNoS_ beta}) Empty

Axiom. (add_SNo_ordinal_eq) We take the following as an axiom:

∀alpha, ordinal alpha → ∀beta, ordinal beta → alpha + beta = SNoCut ({x + beta|x ∈ SNoS_ alpha} ∪ {alpha + x|x ∈ SNoS_ beta}) Empty

Axiom. (add_SNo_ordinal_ordinal) We take the following as an axiom:

∀alpha, ordinal alpha → ∀beta, ordinal beta → ordinal (alpha + beta)

Axiom. (add_SNo_ordinal_SL) We take the following as an axiom:

∀alpha, ordinal alpha → ∀beta, ordinal beta → ordsucc alpha + beta = ordsucc (alpha + beta)

Axiom. (add_SNo_ordinal_SR) We take the following as an axiom:

∀alpha, ordinal alpha → ∀beta, ordinal beta → alpha + ordsucc beta = ordsucc (alpha + beta)

Axiom. (add_SNo_ordinal_InL) We take the following as an axiom:

∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀gamma ∈ alpha, gamma + beta ∈ alpha + beta

Axiom. (add_SNo_ordinal_InR) We take the following as an axiom:

∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀gamma ∈ beta, alpha + gamma ∈ alpha + beta

Axiom. (add_nat_add_SNo) We take the following as an axiom:

∀n m ∈ ω, add_nat n m = n + m

Axiom. (add_SNo_In_omega) We take the following as an axiom:

∀n m ∈ ω, n + m ∈ ω

Axiom. (add_SNo_SNoL_interpolate) We take the following as an axiom:

∀x y, SNo x → SNo y → ∀u ∈ SNoL (x + y), (∃v ∈ SNoL x, u ≤ v + y) ∨ (∃v ∈ SNoL y, u ≤ x + v)

Axiom. (add_SNo_SNoR_interpolate) We take the following as an axiom:

∀x y, SNo x → SNo y → ∀u ∈ SNoR (x + y), (∃v ∈ SNoR x, v + y ≤ u) ∨ (∃v ∈ SNoR y, x + v ≤ u)

Axiom. (add_SNo_assoc) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x + (y + z) = (x + y) + z

Axiom. (add_SNo_cancel_L) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x + y = x + z → y = z

Axiom. (add_SNo_cancel_R) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x + y = z + y → x = z

Axiom. (minus_add_SNo_distr) We take the following as an axiom:

∀x y, SNo x → SNo y → - (x + y) = (- x) + (- y)

Axiom. (minus_add_SNo_distr_3) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → - (x + y + z) = - x + - y + - z

Axiom. (add_SNo_Lev_bd) We take the following as an axiom:

∀x y, SNo x → SNo y → SNoLev (x + y) ⊆ SNoLev x + SNoLev y

Axiom. (add_SNo_SNoS_omega) We take the following as an axiom:

∀x y ∈ SNoS_ ω, x + y ∈ SNoS_ ω

Axiom. (add_SNo_minus_R2) We take the following as an axiom:

∀x y, SNo x → SNo y → (x + y) + - y = x

Axiom. (add_SNo_Lt1_cancel) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x + y < z + y → x < z

Axiom. (add_SNo_Lt2_cancel) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x + y < x + z → y < z

Axiom. (add_SNo_assoc_4) We take the following as an axiom:

∀x y z w, SNo x → SNo y → SNo z → SNo w → x + y + z + w = (x + y + z) + w

Axiom. (add_SNo_com_3_0_1) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x + y + z = y + x + z

Axiom. (add_SNo_com_4_inner_flat) We take the following as an axiom:

∀x y z w, SNo y → SNo z → SNo w → x + y + z + w = x + z + y + w

Axiom. (add_SNo_com_3b_1_2) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → (x + y) + z = (x + z) + y

Axiom. (add_SNo_com_4_inner_mid) We take the following as an axiom:

∀x y z w, SNo x → SNo y → SNo z → SNo w → (x + y) + (z + w) = (x + z) + (y + w)

Axiom. (add_SNo_rotate_3_1) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x + y + z = z + x + y

Axiom. (add_SNo_rotate_4_1) We take the following as an axiom:

∀x y z w, SNo x → SNo y → SNo z → SNo w → x + y + z + w = w + x + y + z

Axiom. (add_SNo_rotate_5_1) We take the following as an axiom:

∀x y z w v, SNo x → SNo y → SNo z → SNo w → SNo v → x + y + z + w + v = v + x + y + z + w

Axiom. (add_SNo_rotate_5_2) We take the following as an axiom:

∀x y z w v, SNo x → SNo y → SNo z → SNo w → SNo v → x + y + z + w + v = w + v + x + y + z

Axiom. (add_SNo_minus_SNo_prop1) We take the following as an axiom:

∀x y, SNo x → SNo y → - x + x + y = y

Axiom. (add_SNo_minus_SNo_prop2) We take the following as an axiom:

∀x y, SNo x → SNo y → x + - x + y = y

Axiom. (add_SNo_minus_SNo_prop3) We take the following as an axiom:

∀x y z w, SNo x → SNo y → SNo z → SNo w → (x + y + z) + (- z + w) = x + y + w

Axiom. (add_SNo_minus_SNo_prop4) We take the following as an axiom:

∀x y z w, SNo x → SNo y → SNo z → SNo w → (x + y + z) + (w + - z) = x + y + w

Axiom. (add_SNo_minus_SNo_prop5) We take the following as an axiom:

∀x y z w, SNo x → SNo y → SNo z → SNo w → (x + y + - z) + (z + w) = x + y + w

Axiom. (add_SNo_minus_Lt1) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x + - y < z → x < z + y

Axiom. (add_SNo_minus_Lt1b) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x < z + y → x + - y < z

Axiom. (add_SNo_minus_Lt2) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → z < x + - y → z + y < x

Axiom. (add_SNo_minus_Lt2b) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → z + y < x → z < x + - y

Axiom. (add_SNo_minus_Lt1b3) We take the following as an axiom:

∀x y z w, SNo x → SNo y → SNo z → SNo w → x + y < w + z → x + y + - z < w

Axiom. (add_SNo_minus_Lt2b3) We take the following as an axiom:

∀x y z w, SNo x → SNo y → SNo z → SNo w → w + z < x + y → w < x + y + - z

Axiom. (add_SNo_Lt_subprop2) We take the following as an axiom:

∀x y z w u v, SNo x → SNo y → SNo z → SNo w → SNo u → SNo v → x + u < z + v → y + v < w + u → x + y < z + w

Axiom. (add_SNo_Lt_subprop3a) We take the following as an axiom:

∀x y z w u a, SNo x → SNo y → SNo z → SNo w → SNo u → SNo a → x + z < w + a → y + a < u → x + y + z < w + u

Axiom. (add_SNo_Lt_subprop3b) We take the following as an axiom:

∀x y w u v a, SNo x → SNo y → SNo w → SNo u → SNo v → SNo a → x + a < w + v → y < a + u → x + y < w + u + v

Axiom. (add_SNo_Lt_subprop3c) We take the following as an axiom:

∀x y z w u a b c, SNo x → SNo y → SNo z → SNo w → SNo u → SNo a → SNo b → SNo c → x + a < b + c → y + c < u → b + z < w + a → x + y + z < w + u

Axiom. (add_SNo_Lt_subprop3d) We take the following as an axiom:

∀x y w u v a b c, SNo x → SNo y → SNo w → SNo u → SNo v → SNo a → SNo b → SNo c → x + a < b + v → y < c + u → b + c < w + a → x + y < w + u + v

Axiom. (ordinal_ordsucc_SNo_eq) We take the following as an axiom:

∀alpha, ordinal alpha → ordsucc alpha = 1 + alpha

Axiom. (add_SNo_omega_eps_Lt) We take the following as an axiom:

∀x y ∈ SNoS_ ω, x < y → ∃n ∈ ω, x + eps_ n < y

Primitive. The name mul_SNo is a term of type set → set → set.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.

Axiom. (mul_SNo_eq) We take the following as an axiom:

∀x, SNo x → ∀y, SNo y → x * y = SNoCut ({(w 0) * y + x * (w 1) + - (w 0) * (w 1)|w ∈ SNoL x ⨯ SNoL y} ∪ {(z 0) * y + x * (z 1) + - (z 0) * (z 1)|z ∈ SNoR x ⨯ SNoR y}) ({(w 0) * y + x * (w 1) + - (w 0) * (w 1)|w ∈ SNoL x ⨯ SNoR y} ∪ {(z 0) * y + x * (z 1) + - (z 0) * (z 1)|z ∈ SNoR x ⨯ SNoL y})

Axiom. (mul_SNo_eq_2) We take the following as an axiom:

∀x y, SNo x → SNo y → ∀p : prop, (∀L R, (∀u, u ∈ L → (∀q : prop, (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, u = w0 * y + x * w1 + - w0 * w1 → q) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, u = z0 * y + x * z1 + - z0 * z1 → q) → q)) → (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, w0 * y + x * w1 + - w0 * w1 ∈ L) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, z0 * y + x * z1 + - z0 * z1 ∈ L) → (∀u, u ∈ R → (∀q : prop, (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, u = w0 * y + x * z1 + - w0 * z1 → q) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, u = z0 * y + x * w1 + - z0 * w1 → q) → q)) → (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, w0 * y + x * z1 + - w0 * z1 ∈ R) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, z0 * y + x * w1 + - z0 * w1 ∈ R) → x * y = SNoCut L R → p) → p

Axiom. (mul_SNo_prop_1) We take the following as an axiom:

∀x, SNo x → ∀y, SNo y → ∀p : prop, (SNo (x * y) → (∀u ∈ SNoL x, ∀v ∈ SNoL y, u * y + x * v < x * y + u * v) → (∀u ∈ SNoR x, ∀v ∈ SNoR y, u * y + x * v < x * y + u * v) → (∀u ∈ SNoL x, ∀v ∈ SNoR y, x * y + u * v < u * y + x * v) → (∀u ∈ SNoR x, ∀v ∈ SNoL y, x * y + u * v < u * y + x * v) → p) → p

Axiom. (SNo_mul_SNo) We take the following as an axiom:

∀x y, SNo x → SNo y → SNo (x * y)

Axiom. (mul_SNo_eq_3) We take the following as an axiom:

∀x y, SNo x → SNo y → ∀p : prop, (∀L R, SNoCutP L R → (∀u, u ∈ L → (∀q : prop, (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, u = w0 * y + x * w1 + - w0 * w1 → q) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, u = z0 * y + x * z1 + - z0 * z1 → q) → q)) → (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, w0 * y + x * w1 + - w0 * w1 ∈ L) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, z0 * y + x * z1 + - z0 * z1 ∈ L) → (∀u, u ∈ R → (∀q : prop, (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, u = w0 * y + x * z1 + - w0 * z1 → q) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, u = z0 * y + x * w1 + - z0 * w1 → q) → q)) → (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, w0 * y + x * z1 + - w0 * z1 ∈ R) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, z0 * y + x * w1 + - z0 * w1 ∈ R) → x * y = SNoCut L R → p) → p

Axiom. (mul_SNo_Lt) We take the following as an axiom:

∀x y u v, SNo x → SNo y → SNo u → SNo v → u < x → v < y → u * y + x * v < x * y + u * v

Axiom. (mul_SNo_Le) We take the following as an axiom:

∀x y u v, SNo x → SNo y → SNo u → SNo v → u ≤ x → v ≤ y → u * y + x * v ≤ x * y + u * v

Axiom. (mul_SNo_SNoL_interpolate) We take the following as an axiom:

∀x y, SNo x → SNo y → ∀u ∈ SNoL (x * y), (∃v ∈ SNoL x, ∃w ∈ SNoL y, u + v * w ≤ v * y + x * w) ∨ (∃v ∈ SNoR x, ∃w ∈ SNoR y, u + v * w ≤ v * y + x * w)

Axiom. (mul_SNo_SNoL_interpolate_impred) We take the following as an axiom:

∀x y, SNo x → SNo y → ∀u ∈ SNoL (x * y), ∀p : prop, (∀v ∈ SNoL x, ∀w ∈ SNoL y, u + v * w ≤ v * y + x * w → p) → (∀v ∈ SNoR x, ∀w ∈ SNoR y, u + v * w ≤ v * y + x * w → p) → p

Axiom. (mul_SNo_SNoR_interpolate) We take the following as an axiom:

∀x y, SNo x → SNo y → ∀u ∈ SNoR (x * y), (∃v ∈ SNoL x, ∃w ∈ SNoR y, v * y + x * w ≤ u + v * w) ∨ (∃v ∈ SNoR x, ∃w ∈ SNoL y, v * y + x * w ≤ u + v * w)

Axiom. (mul_SNo_SNoR_interpolate_impred) We take the following as an axiom:

∀x y, SNo x → SNo y → ∀u ∈ SNoR (x * y), ∀p : prop, (∀v ∈ SNoL x, ∀w ∈ SNoR y, v * y + x * w ≤ u + v * w → p) → (∀v ∈ SNoR x, ∀w ∈ SNoL y, v * y + x * w ≤ u + v * w → p) → p

Axiom. (mul_SNo_zeroR) We take the following as an axiom:

∀x, SNo x → x * 0 = 0

Axiom. (mul_SNo_oneR) We take the following as an axiom:

∀x, SNo x → x * 1 = x

Axiom. (mul_SNo_com) We take the following as an axiom:

∀x y, SNo x → SNo y → x * y = y * x

Axiom. (mul_SNo_minus_distr) We take the following as an axiom:

∀x y, SNo x → SNo y → (- x) * y = - x * y

Axiom. (mul_SNo_distrR) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → (x + y) * z = x * z + y * z

Axiom. (mul_SNo_distrL) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x * (y + z) = x * y + x * z

Axiom. (mul_SNo_assoc) We take the following as an axiom:

∀x y z, SNo x → SNo y → SNo z → x * (y * z) = (x * y) * z

Axiom. (mul_nat_mul_SNo) We take the following as an axiom:

∀n m ∈ ω, mul_nat n m = n * m

Axiom. (mul_SNo_In_omega) We take the following as an axiom:

∀n m ∈ ω, n * m ∈ ω

Definition. We define diadic_open to be λX ⇒ X ⊆ SNoS_ ω ∧ ∀x ∈ X, ∃n ∈ ω, ∀y ∈ SNoS_ ω, x + - eps_ n < y → y < x + eps_ n → y ∈ X of type set → prop.

Axiom. (diadic_open_I) We take the following as an axiom:

∀X ⊆ SNoS_ ω, (∀x ∈ X, ∃n ∈ ω, ∀y ∈ SNoS_ ω, x + - eps_ n < y → y < x + eps_ n → y ∈ X) → diadic_open X

Definition. We define SNoL_omega to be λx ⇒ {y ∈ SNoS_ ω|y < x} of type set → set.

Definition. We define SNoR_omega to be λx ⇒ {y ∈ SNoS_ ω|x < y} of type set → set.

Axiom. (diadic_open_SNoL_omega_I) We take the following as an axiom:

∀z, SNo z → (∀x ∈ SNoL_omega z, ∃n ∈ ω, x + eps_ n < z) → diadic_open (SNoL_omega z)

Axiom. (diadic_open_SNoR_omega_I) We take the following as an axiom:

∀z, SNo z → (∀x ∈ SNoR_omega z, ∃n ∈ ω, z < x + - eps_ n) → diadic_open (SNoR_omega z)

Definition. We define real to be {x ∈ SNoS_ (ordsucc ω)|SNoL_omega x ≠ 0 ∧ SNoR_omega x ≠ 0 ∧ diadic_open (SNoL_omega x) ∧ diadic_open (SNoR_omega x)} of type set.

Axiom. (real_I) We take the following as an axiom:

∀x ∈ SNoS_ (ordsucc ω), SNoL_omega x ≠ 0 → SNoR_omega x ≠ 0 → diadic_open (SNoL_omega x) → diadic_open (SNoR_omega x) → x ∈ real

Axiom. (real_E) We take the following as an axiom:

∀x ∈ real, ∀p : prop, (x ∈ SNoS_ (ordsucc ω) → SNoL_omega x ≠ 0 → SNoR_omega x ≠ 0 → diadic_open (SNoL_omega x) → diadic_open (SNoR_omega x) → p) → p

Axiom. (Subq_real_SNoS_ordsucc_omega) We take the following as an axiom:

real ⊆ SNoS_ (ordsucc ω)

Axiom. (Subq_SNoS_omega_real) We take the following as an axiom:

∀x ∈ SNoS_ ω, x ∈ real

Axiom. (SNoCutP_SNoL_SNoR_omega) We take the following as an axiom:

∀x, SNo x → SNoCutP (SNoL_omega x) (SNoR_omega x)

Axiom. (SNoS_ordsucc_omega_SNoL_SNoR_omega) We take the following as an axiom:

∀x ∈ SNoS_ (ordsucc ω), x = SNoCut (SNoL_omega x) (SNoR_omega x)

Axiom. (real_SNoL_SNoR_omega) We take the following as an axiom:

∀x ∈ real, x = SNoCut (SNoL_omega x) (SNoR_omega x)

Axiom. (real_ex_diad_Lt) We take the following as an axiom:

∀x ∈ real, ∃w ∈ SNoS_ ω, w < x

Axiom. (real_ex_diad_Gt) We take the following as an axiom:

∀x ∈ real, ∃z ∈ SNoS_ ω, x < z

Axiom. (mul_SNo_pos_pos) We take the following as an axiom:

∀x y, SNo x → SNo y → 0 < x → 0 < y → 0 < x * y

Axiom. (mul_SNo_pos_neg) We take the following as an axiom:

∀x y, SNo x → SNo y → 0 < x → y < 0 → x * y < 0

Axiom. (mul_SNo_neg_pos) We take the following as an axiom:

∀x y, SNo x → SNo y → x < 0 → 0 < y → x * y < 0

Axiom. (mul_SNo_neg_neg) We take the following as an axiom:

∀x y, SNo x → SNo y → x < 0 → y < 0 → 0 < x * y

Axiom. (mul_SNo_nonzero) We take the following as an axiom:

∀x y, SNo x → SNo y → x ≠ 0 → y ≠ 0 → x * y ≠ 0

Axiom. (minus_SNo_restr_SNo) We take the following as an axiom:

∀x, SNo x → ∀alpha ∈ SNoLev x, (- x) ∩ SNoElts_ alpha = - (x ∩ SNoElts_ alpha)

Axiom. (minus_SNo_exactly1of2) We take the following as an axiom:

∀x, SNo x → ∀alpha ∈ SNoLev x, exactly1of2 (alpha ∈ x) (alpha ∈ - x)

Axiom. (minus_SNo_In) We take the following as an axiom:

∀x, SNo x → ∀alpha ∈ SNoLev x, alpha ∈ x → alpha ∉ - x

Axiom. (minus_SNo_nIn) We take the following as an axiom:

∀x, SNo x → ∀alpha ∈ SNoLev x, alpha ∉ x → alpha ∈ - x

Axiom. (real_minus_SNo) We take the following as an axiom:

∀x ∈ real, - x ∈ real

Definition. We define div_SNo to be λx y ⇒ if y = 0 then 0 else Eps_i (λz ⇒ SNo z ∧ z * y = x) of type set → set → set.

Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.

Definition. We define exp_SNo_nat to be λn m : set ⇒ nat_primrec 1 (λ_ r ⇒ n * r) m of type set → set → set.

Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.

End of Section SurrealArithmetic

Definition. We define CSNo to be λz ⇒ ∃x, SNo x ∧ ∃y, SNo y ∧ z = SNo_pair x y of type set → prop.

Axiom. (CSNo_I) We take the following as an axiom:

∀x y, SNo x → SNo y → CSNo (SNo_pair x y)

Axiom. (CSNo_E) We take the following as an axiom:

∀z, CSNo z → ∀p : set → prop, (∀x y, SNo x → SNo y → z = SNo_pair x y → p (SNo_pair x y)) → p z

Axiom. (SNo_CSNo) We take the following as an axiom:

∀x, SNo x → CSNo x

Beginning of Section Complex

Definition. We define Complex_i to be SNo_pair 0 1 of type set.

Let i ≝ Complex_i

Axiom. (SNo_Complex_i) We take the following as an axiom:

CSNo i

Definition. We define CSNo_Re to be λz ⇒ Eps_i (λx ⇒ SNo x ∧ ∃y, SNo y ∧ z = SNo_pair x y) of type set → set.

Definition. We define CSNo_Im to be λz ⇒ Eps_i (λy ⇒ SNo y ∧ z = SNo_pair (CSNo_Re z) y) of type set → set.

Let Re : set → set ≝ CSNo_Re

Let Im : set → set ≝ CSNo_Im

Let pa : set → set → set ≝ SNo_pair

Axiom. (CSNo_Re1) We take the following as an axiom:

∀z, CSNo z → SNo (Re z) ∧ ∃y, SNo y ∧ z = pa (Re z) y

Axiom. (CSNo_Re2) We take the following as an axiom:

∀x y, SNo x → SNo y → Re (pa x y) = x

Axiom. (CSNo_Im1) We take the following as an axiom:

∀z, CSNo z → SNo (Im z) ∧ z = pa (Re z) (Im z)

Axiom. (CSNo_Im2) We take the following as an axiom:

∀x y, SNo x → SNo y → Im (pa x y) = y

Axiom. (CSNo_ReR) We take the following as an axiom:

∀z, CSNo z → SNo (Re z)

Axiom. (CSNo_ImR) We take the following as an axiom:

∀z, CSNo z → SNo (Im z)

Axiom. (CSNo_ReIm) We take the following as an axiom:

∀z, CSNo z → z = pa (Re z) (Im z)

Axiom. (CSNo_ReIm_split) We take the following as an axiom:

∀z w, CSNo z → CSNo w → Re z = Re w → Im z = Im w → z = w

Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.

Definition. We define minus_CSNo to be λz ⇒ pa (- Re z) (- Im z) of type set → set.

Definition. We define add_CSNo to be λz w ⇒ pa (Re z + Re w) (Im z + Im w) of type set → set → set.

Definition. We define mul_CSNo to be λz w ⇒ pa (Re z * Re w + - (Im z * Im w)) (Re z * Im w + Im z * Re w) of type set → set → set.

Definition. We define div_CSNo to be λx y ⇒ if y = 0 then 0 else Eps_i (λz ⇒ CSNo z ∧ mul_CSNo z y = x) of type set → set → set.

Axiom. (CSNo_minus_CSNo) We take the following as an axiom:

∀z, CSNo z → CSNo (minus_CSNo z)

Axiom. (SNo_Re) We take the following as an axiom:

∀x, SNo x → Re x = x

Axiom. (SNo_Im) We take the following as an axiom:

∀x, SNo x → Im x = 0

Axiom. (Re_0) We take the following as an axiom:

Re 0 = 0

Axiom. (Im_0) We take the following as an axiom:

Im 0 = 0

Axiom. (Re_1) We take the following as an axiom:

Re 1 = 1

Axiom. (Im_1) We take the following as an axiom:

Im 1 = 0

Axiom. (Re_i) We take the following as an axiom:

Re i = 0

Axiom. (Im_i) We take the following as an axiom:

Im i = 1

Axiom. (add_SNo_add_CSNo) We take the following as an axiom:

∀x y, SNo x → SNo y → x + y = add_CSNo x y

Axiom. (CSNo_add_CSNo) We take the following as an axiom:

∀z w, CSNo z → CSNo w → CSNo (add_CSNo z w)

Axiom. (add_CSNo_0L) We take the following as an axiom:

∀z, CSNo z → add_CSNo 0 z = z

Axiom. (add_CSNo_0R) We take the following as an axiom:

∀z, CSNo z → add_CSNo z 0 = z

Axiom. (add_CSNo_minus_CSNo_linv) We take the following as an axiom:

∀z, CSNo z → add_CSNo (minus_CSNo z) z = 0

Axiom. (add_CSNo_minus_CSNo_rinv) We take the following as an axiom:

∀z, CSNo z → add_CSNo z (minus_CSNo z) = 0

Axiom. (minus_SNo_minus_CSNo) We take the following as an axiom:

∀x, SNo x → - x = minus_CSNo x

End of Section Complex

Beginning of Section Complex

Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_CSNo.

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_CSNo.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_CSNo.

Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_CSNo.

Definition. We define int to be ω ∪ {- n|n ∈ ω} of type set.

Definition. We define rational to be ReplSep2 int (λ_ ⇒ ω) (λnum den ⇒ den ≠ 0) (λnum den ⇒ num :/: den) of type set.

Definition. We define Sum to be λm n f ⇒ nat_primrec 0 (λk r ⇒ if k ∈ m then 0 else f k + r) (ordsucc n) of type set → set → (set → set) → set.

Definition. We define Prod to be λm n f ⇒ nat_primrec 1 (λk r ⇒ if k ∈ m then 1 else f k * r) (ordsucc n) of type set → set → (set → set) → set.

End of Section Complex

Beginning of Section Int

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.

Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.

Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.

Axiom. (int_SNo_cases) We take the following as an axiom:

∀p : set → prop, (∀n ∈ ω, p n) → (∀n ∈ ω, p (- n)) → ∀x ∈ int, p x

Axiom. (Subq_omega_int) We take the following as an axiom:

ω ⊆ int

Axiom. (int_minus_SNo_omega) We take the following as an axiom:

∀n ∈ ω, - n ∈ int

Axiom. (int_minus_SNo) We take the following as an axiom:

∀x ∈ int, - x ∈ int

Axiom. (int_add_SNo_lem) We take the following as an axiom:

∀n ∈ ω, ∀m, nat_p m → - n + m ∈ int

Axiom. (int_add_SNo) We take the following as an axiom:

∀x y ∈ int, x + y ∈ int

Axiom. (int_mul_SNo) We take the following as an axiom:

∀x y ∈ int, x * y ∈ int

Definition. We define divides_int to be λm n ⇒ m ∈ int ∧ n ∈ int ∧ ∃k ∈ int, m * k = n of type set → set → prop.

Definition. We define equiv_int_mod to be λm k n ⇒ m ∈ int ∧ k ∈ int ∧ n ∈ ω ∖ 1 ∧ divides_int (m + - k) n of type set → set → set → prop.

Definition. We define coprime_int to be λa b ⇒ a ∈ int ∧ b ∈ int ∧ ∀x ∈ ω ∖ 1, divides_int x a → divides_int x b → x = 1 of type set → set → prop.

End of Section Int

Primitive. The name pack_b_b_r_e_e is a term of type set → (set → set → set) → (set → set → set) → (set → set → prop) → set → set → set.

Axiom. (pack_b_b_r_e_e_0_eq) We take the following as an axiom:

∀S X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, S = pack_b_b_r_e_e X f g R c d → X = S 0

Axiom. (pack_b_b_r_e_e_0_eq2) We take the following as an axiom:

∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, X = pack_b_b_r_e_e X f g R c d 0

Axiom. (pack_b_b_r_e_e_1_eq) We take the following as an axiom:

∀S X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, S = pack_b_b_r_e_e X f g R c d → ∀x y ∈ X, f x y = decode_b (S 1) x y

Axiom. (pack_b_b_r_e_e_1_eq2) We take the following as an axiom:

∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, ∀x y ∈ X, f x y = decode_b (pack_b_b_r_e_e X f g R c d 1) x y

Axiom. (pack_b_b_r_e_e_2_eq) We take the following as an axiom:

∀S X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, S = pack_b_b_r_e_e X f g R c d → ∀x y ∈ X, g x y = decode_b (S 2) x y

Axiom. (pack_b_b_r_e_e_2_eq2) We take the following as an axiom:

∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, ∀x y ∈ X, g x y = decode_b (pack_b_b_r_e_e X f g R c d 2) x y

Axiom. (pack_b_b_r_e_e_3_eq) We take the following as an axiom:

∀S X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, S = pack_b_b_r_e_e X f g R c d → ∀x y ∈ X, R x y = decode_r (S 3) x y

Axiom. (pack_b_b_r_e_e_3_eq2) We take the following as an axiom:

∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, ∀x y ∈ X, R x y = decode_r (pack_b_b_r_e_e X f g R c d 3) x y

Axiom. (pack_b_b_r_e_e_4_eq) We take the following as an axiom:

∀S X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, S = pack_b_b_r_e_e X f g R c d → c = S 4

Axiom. (pack_b_b_r_e_e_4_eq2) We take the following as an axiom:

∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, c = pack_b_b_r_e_e X f g R c d 4

Axiom. (pack_b_b_r_e_e_5_eq) We take the following as an axiom:

∀S X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, S = pack_b_b_r_e_e X f g R c d → d = S 5

Axiom. (pack_b_b_r_e_e_5_eq2) We take the following as an axiom:

∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, d = pack_b_b_r_e_e X f g R c d 5

Axiom. (pack_b_b_r_e_e_inj) We take the following as an axiom:

∀X X', ∀f f' : set → set → set, ∀g g' : set → set → set, ∀R R' : set → set → prop, ∀c c' : set, ∀d d' : set, pack_b_b_r_e_e X f g R c d = pack_b_b_r_e_e X' f' g' R' c' d' → X = X' ∧ (∀x y ∈ X, f x y = f' x y) ∧ (∀x y ∈ X, g x y = g' x y) ∧ (∀x y ∈ X, R x y = R' x y) ∧ c = c' ∧ d = d'

Axiom. (pack_b_b_r_e_e_ext) We take the following as an axiom:

∀X, ∀f f' : set → set → set, ∀g g' : set → set → set, ∀R R' : set → set → prop, ∀c, ∀d, (∀x y ∈ X, f x y = f' x y) → (∀x y ∈ X, g x y = g' x y) → (∀x y ∈ X, R x y ↔ R' x y) → pack_b_b_r_e_e X f g R c d = pack_b_b_r_e_e X f' g' R' c d

Definition. We define struct_b_b_r_e_e to be λS ⇒ ∀q : set → prop, (∀X : set, ∀f : set → set → set, (∀x y ∈ X, f x y ∈ X) → ∀g : set → set → set, (∀x y ∈ X, g x y ∈ X) → ∀R : set → set → prop, ∀c : set, c ∈ X → ∀d : set, d ∈ X → q (pack_b_b_r_e_e X f g R c d)) → q S of type set → prop.

Axiom. (pack_struct_b_b_r_e_e_I) We take the following as an axiom:

∀X, ∀f : set → set → set, (∀x y ∈ X, f x y ∈ X) → ∀g : set → set → set, (∀x y ∈ X, g x y ∈ X) → ∀R : set → set → prop, ∀c : set, c ∈ X → ∀d : set, d ∈ X → struct_b_b_r_e_e (pack_b_b_r_e_e X f g R c d)

Axiom. (pack_struct_b_b_r_e_e_E1) We take the following as an axiom:

∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, struct_b_b_r_e_e (pack_b_b_r_e_e X f g R c d) → ∀x y ∈ X, f x y ∈ X

Axiom. (pack_struct_b_b_r_e_e_E2) We take the following as an axiom:

∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, struct_b_b_r_e_e (pack_b_b_r_e_e X f g R c d) → ∀x y ∈ X, g x y ∈ X

Axiom. (pack_struct_b_b_r_e_e_E4) We take the following as an axiom:

∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, struct_b_b_r_e_e (pack_b_b_r_e_e X f g R c d) → c ∈ X

Axiom. (pack_struct_b_b_r_e_e_E5) We take the following as an axiom:

∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, struct_b_b_r_e_e (pack_b_b_r_e_e X f g R c d) → d ∈ X

Axiom. (struct_b_b_r_e_e_eta) We take the following as an axiom:

∀S, struct_b_b_r_e_e S → S = pack_b_b_r_e_e (S 0) (decode_b (S 1)) (decode_b (S 2)) (decode_r (S 3)) (S 4) (S 5)

Primitive. The name unpack_b_b_r_e_e_i is a term of type set → (set → (set → set → set) → (set → set → set) → (set → set → prop) → set → set → set) → set.

Axiom. (unpack_b_b_r_e_e_i_eq) We take the following as an axiom:

∀Phi : set → (set → set → set) → (set → set → set) → (set → set → prop) → set → set → set, ∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, (∀f' : set → set → set, (∀x y ∈ X, f x y = f' x y) → ∀g' : set → set → set, (∀x y ∈ X, g x y = g' x y) → ∀R' : set → set → prop, (∀x y ∈ X, R x y ↔ R' x y) → Phi X f' g' R' c d = Phi X f g R c d) → unpack_b_b_r_e_e_i (pack_b_b_r_e_e X f g R c d) Phi = Phi X f g R c d

Primitive. The name unpack_b_b_r_e_e_o is a term of type set → (set → (set → set → set) → (set → set → set) → (set → set → prop) → set → set → prop) → prop.

Axiom. (unpack_b_b_r_e_e_o_eq) We take the following as an axiom:

∀Phi : set → (set → set → set) → (set → set → set) → (set → set → prop) → set → set → prop, ∀X, ∀f : set → set → set, ∀g : set → set → set, ∀R : set → set → prop, ∀c : set, ∀d : set, (∀f' : set → set → set, (∀x y ∈ X, f x y = f' x y) → ∀g' : set → set → set, (∀x y ∈ X, g x y = g' x y) → ∀R' : set → set → prop, (∀x y ∈ X, R x y ↔ R' x y) → Phi X f' g' R' c d = Phi X f g R c d) → unpack_b_b_r_e_e_o (pack_b_b_r_e_e X f g R c d) Phi = Phi X f g R c d

Primitive. The name OrderedFieldStruct is a term of type set → prop.

Beginning of Section explicit_OrderedField_RepIndep2

Variable R : set

Variable zero one : set

Variable plus mult : set → set → set

Variable leq : set → set → prop

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Variable plus' mult' : set → set → set

Variable leq' : set → set → prop

Notation. We use + as an infix operator with priority 355 and which associates to the right corresponding to applying term plus'.

Notation. We use ⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term mult'.

Hypothesis Hpp' : ∀a b ∈ R, a + b = a + b

Hypothesis Hmm' : ∀a b ∈ R, a * b = a ⨯ b

Hypothesis Hll' : ∀a b ∈ R, leq a b ↔ leq' a b

Axiom. (explicit_OrderedField_repindep) We take the following as an axiom:

explicit_OrderedField R zero one plus mult leq ↔ explicit_OrderedField R zero one plus' mult' leq'

End of Section explicit_OrderedField_RepIndep2

Axiom. (OrderedFieldStruct_unpack_eq) We take the following as an axiom:

∀R, ∀plus mult : set → set → set, ∀leq : set → set → prop, ∀zero one, unpack_b_b_r_e_e_o (pack_b_b_r_e_e R plus mult leq zero one) (λR plus mult leq zero one ⇒ explicit_OrderedField R zero one plus mult leq) = explicit_OrderedField R zero one plus mult leq

Definition. We define RealsStruct to be λR ⇒ struct_b_b_r_e_e R ∧ unpack_b_b_r_e_e_o R (λR plus mult leq zero one ⇒ explicit_Reals R zero one plus mult leq) of type set → prop.

Beginning of Section explicit_Reals_RepIndep2

Variable R : set

Variable zero one : set

Variable plus mult : set → set → set

Variable leq : set → set → prop

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term plus.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mult.

Variable plus' mult' : set → set → set

Variable leq' : set → set → prop

Notation. We use + as an infix operator with priority 355 and which associates to the right corresponding to applying term plus'.

Notation. We use ⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term mult'.

Hypothesis Hpp' : ∀a b ∈ R, a + b = a + b

Hypothesis Hmm' : ∀a b ∈ R, a * b = a ⨯ b

Hypothesis Hll' : ∀a b ∈ R, leq a b ↔ leq' a b

Axiom. (explicit_Reals_repindep) We take the following as an axiom:

explicit_Reals R zero one plus mult leq ↔ explicit_Reals R zero one plus' mult' leq'

End of Section explicit_Reals_RepIndep2

Axiom. (RealsStruct_unpack_eq) We take the following as an axiom:

∀R, ∀plus mult : set → set → set, ∀leq : set → set → prop, ∀zero one, unpack_b_b_r_e_e_o (pack_b_b_r_e_e R plus mult leq zero one) (λR plus mult leq zero one ⇒ explicit_Reals R zero one plus mult leq) = explicit_Reals R zero one plus mult leq

Definition. We define RealsStruct_carrier to be λRs ⇒ Rs 0 of type set → set.

Definition. We define RealsStruct_plus to be λRs ⇒ decode_b (Rs 1) of type set → set → set → set.

Definition. We define RealsStruct_mult to be λRs ⇒ decode_b (Rs 2) of type set → set → set → set.

Definition. We define RealsStruct_leq to be λRs ⇒ decode_r (Rs 3) of type set → set → set → prop.

Definition. We define RealsStruct_zero to be λRs ⇒ Rs 4 of type set → set.

Definition. We define RealsStruct_one to be λRs ⇒ Rs 5 of type set → set.

Primitive. The name Field_of_RealsStruct is a term of type set → set.

Axiom. (Field_of_RealsStruct_0) We take the following as an axiom:

∀Rs, Field_of_RealsStruct Rs 0 = RealsStruct_carrier Rs

Axiom. (Field_of_RealsStruct_1) We take the following as an axiom:

∀Rs, ∀x y ∈ RealsStruct_carrier Rs, Field_of_RealsStruct Rs 1 x y = RealsStruct_plus Rs x y

Axiom. (Field_of_RealsStruct_2) We take the following as an axiom:

∀Rs, ∀x y ∈ RealsStruct_carrier Rs, Field_of_RealsStruct Rs 2 x y = RealsStruct_mult Rs x y

Axiom. (Field_of_RealsStruct_3) We take the following as an axiom:

∀Rs, Field_of_RealsStruct Rs 3 = RealsStruct_zero Rs

Axiom. (Field_of_RealsStruct_4) We take the following as an axiom:

∀Rs, Field_of_RealsStruct Rs 4 = RealsStruct_one Rs

Beginning of Section RealsStruct

Variable Rs : set

Hypothesis HRs : RealsStruct Rs

Let R : set ≝ RealsStruct_carrier Rs

Let zero : set ≝ RealsStruct_zero Rs

Let one : set ≝ RealsStruct_one Rs

Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term RealsStruct_plus Rs.

Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term RealsStruct_mult Rs.

Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term RealsStruct_leq Rs.

Axiom. (RealsStruct_eta) We take the following as an axiom:

Rs = pack_b_b_r_e_e R (RealsStruct_plus Rs) (RealsStruct_mult Rs) (RealsStruct_leq Rs) zero one

Axiom. (RealsStruct_explicit_Reals) We take the following as an axiom:

explicit_Reals R zero one (RealsStruct_plus Rs) (RealsStruct_mult Rs) (RealsStruct_leq Rs)

Axiom. (Field_of_RealsStruct_is_CRing_with_id) We take the following as an axiom:

CRing_with_id (Field_of_RealsStruct Rs)

Definition. We define RealsStruct_lt to be λx y ⇒ x ≤ y ∧ x ≠ y of type set → set → prop.

Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term RealsStruct_lt.

Axiom. (explicit_Field_of_RealsStruct) We take the following as an axiom:

explicit_Field R zero one (RealsStruct_plus Rs) (RealsStruct_mult Rs)

Axiom. (explicit_OrderedField_of_RealsStruct) We take the following as an axiom:

explicit_OrderedField R zero one (RealsStruct_plus Rs) (RealsStruct_mult Rs) (RealsStruct_leq Rs)

Axiom. (RealsStruct_OrderedField) We take the following as an axiom:

OrderedFieldStruct Rs

Axiom. (Field_of_RealsStruct_1f) We take the following as an axiom:

(λx y : set ⇒ Field_of_RealsStruct Rs 1 x y) = RealsStruct_plus Rs

Axiom. (Field_of_RealsStruct_2f) We take the following as an axiom:

(λx y : set ⇒ Field_of_RealsStruct Rs 2 x y) = RealsStruct_mult Rs

Axiom. (explicit_Field_of_RealsStruct_2) We take the following as an axiom:

explicit_Field R (Field_of_RealsStruct Rs 3) (Field_of_RealsStruct Rs 4) (decode_b (Field_of_RealsStruct Rs 1)) (decode_b (Field_of_RealsStruct Rs 2))

Axiom. (Field_Field_of_RealsStruct) We take the following as an axiom:

Field (Field_of_RealsStruct Rs)

Axiom. (RealsStruct_zero_In) We take the following as an axiom:

zero ∈ R

Axiom. (RealsStruct_one_In) We take the following as an axiom:

one ∈ R

Axiom. (RealsStruct_plus_clos) We take the following as an axiom:

∀x y ∈ R, x + y ∈ R

Axiom. (RealsStruct_mult_clos) We take the following as an axiom:

∀x y ∈ R, x * y ∈ R

Axiom. (RealsStruct_plus_assoc) We take the following as an axiom:

∀x y z ∈ R, x + (y + z) = (x + y) + z

Axiom. (RealsStruct_plus_com) We take the following as an axiom:

∀x y ∈ R, x + y = y + x

Axiom. (RealsStruct_zero_L) We take the following as an axiom:

∀x ∈ R, zero + x = x

Axiom. (RealsStruct_mult_assoc) We take the following as an axiom:

∀x y z ∈ R, x * (y * z) = (x * y) * z

Axiom. (RealsStruct_mult_com) We take the following as an axiom:

∀x y ∈ R, x * y = y * x

Axiom. (RealsStruct_one_neq_zero) We take the following as an axiom:

one ≠ zero

Axiom. (RealsStruct_one_L) We take the following as an axiom:

∀x ∈ R, one * x = x

Axiom. (RealsStruct_distr_L) We take the following as an axiom:

∀x y z ∈ R, x * (y + z) = x * y + x * z

Axiom. (RealsStruct_leq_refl) We take the following as an axiom:

∀x ∈ R, x ≤ x

Axiom. (RealsStruct_leq_tra) We take the following as an axiom:

∀x y z ∈ R, x ≤ y → y ≤ z → x ≤ z

Axiom. (RealsStruct_leq_antisym) We take the following as an axiom:

∀x y ∈ R, x ≤ y → y ≤ x → x = y

Axiom. (RealsStruct_leq_linear) We take the following as an axiom:

∀x y ∈ R, x ≤ y ∨ y ≤ x

Axiom. (RealsStruct_leq_plus) We take the following as an axiom:

∀x y z ∈ R, x ≤ y → x + z ≤ y + z

Axiom. (RealsStruct_lt_leq) We take the following as an axiom:

∀x y ∈ R, x < y → x ≤ y

Axiom. (RealsStruct_lt_irref) We take the following as an axiom:

∀x ∈ R, ¬ (x < x)

Axiom. (RealsStruct_lt_leq_asym) We take the following as an axiom:

∀x y ∈ R, x < y → ¬ (y ≤ x)

Axiom. (RealsStruct_leq_lt_asym) We take the following as an axiom:

∀x y ∈ R, x ≤ y → ¬ (y < x)

Axiom. (RealsStruct_lt_asym) We take the following as an axiom:

∀x y ∈ R, x < y → ¬ (y < x)

Axiom. (RealsStruct_lt_leq_tra) We take the following as an axiom:

∀x y z ∈ R, x < y → y ≤ z → x < z

Axiom. (RealsStruct_leq_lt_tra) We take the following as an axiom:

∀x y z ∈ R, x ≤ y → y < z → x < z

Axiom. (RealsStruct_lt_tra) We take the following as an axiom:

∀x y z ∈ R, x < y → y < z → x < z

Axiom. (RealsStruct_lt_trich_impred) We take the following as an axiom:

∀x y ∈ R, ∀p : prop, (x < y → p) → (x = y → p) → (y < x → p) → p

Axiom. (RealsStruct_lt_trich) We take the following as an axiom:

∀x y ∈ R, x < y ∨ x = y ∨ y < x

Axiom. (RealsStruct_leq_lt_linear) We take the following as an axiom:

∀x y ∈ R, x ≤ y ∨ y < x

Notation. We use - as a prefix operator with priority 358 corresponding to applying term Field_minus (Field_of_RealsStruct Rs).

Axiom. (RealsStruct_minus_eq) We take the following as an axiom:

∀x ∈ R, - x = explicit_Field_minus R (Field_of_RealsStruct Rs 3) (Field_of_RealsStruct Rs 4) (decode_b (Field_of_RealsStruct Rs 1)) (decode_b (Field_of_RealsStruct Rs 2)) x

Axiom. (RealsStruct_minus_clos) We take the following as an axiom:

∀x ∈ R, - x ∈ R

Axiom. (RealsStruct_minus_R) We take the following as an axiom:

∀x ∈ R, x + - x = zero

Axiom. (RealsStruct_minus_L) We take the following as an axiom:

∀x ∈ R, - x + x = zero

Axiom. (RealsStruct_plus_cancelL) We take the following as an axiom:

∀x y z ∈ R, x + y = x + z → y = z

Axiom. (RealsStruct_minus_eq2) We take the following as an axiom:

∀x ∈ R, - x = explicit_Field_minus R zero one (RealsStruct_plus Rs) (RealsStruct_mult Rs) x

Axiom. (RealsStruct_plus_cancelR) We take the following as an axiom:

∀x y z ∈ R, x + z = y + z → x = y

Axiom. (RealsStruct_minus_invol) We take the following as an axiom:

∀x ∈ R, - - x = x

Axiom. (RealsStruct_minus_one_In) We take the following as an axiom:

- one ∈ R

Axiom. (RealsStruct_zero_multR) We take the following as an axiom:

∀x ∈ R, x * zero = zero

Axiom. (RealsStruct_zero_multL) We take the following as an axiom:

∀x ∈ R, zero * x = zero

Axiom. (RealsStruct_minus_mult) We take the following as an axiom:

∀x ∈ R, - x = (- one) * x

Axiom. (RealsStruct_minus_one_square) We take the following as an axiom:

(- one) * (- one) = one

Axiom. (RealsStruct_minus_square) We take the following as an axiom:

∀x ∈ R, (- x) * (- x) = x * x

Axiom. (RealsStruct_minus_zero) We take the following as an axiom:

- zero = zero

Axiom. (RealsStruct_dist_R) We take the following as an axiom:

∀x y z ∈ R, (x + y) * z = x * z + y * z

Axiom. (RealsStruct_minus_plus_dist) We take the following as an axiom:

∀x y ∈ R, - (x + y) = - x + - y

Axiom. (RealsStruct_minus_mult_L) We take the following as an axiom:

∀x y ∈ R, (- x) * y = - (x * y)

Axiom. (RealsStruct_minus_mult_R) We take the following as an axiom:

∀x y ∈ R, x * (- y) = - (x * y)

Axiom. (RealsStruct_mult_zero_inv) We take the following as an axiom:

∀x y ∈ R, x * y = zero → x = zero ∨ y = zero

Axiom. (RealsStruct_square_zero_inv) We take the following as an axiom:

∀x ∈ R, x * x = zero → x = zero

Axiom. (RealsStruct_minus_leq) We take the following as an axiom:

∀x y ∈ R, x ≤ y → - y ≤ - x

Axiom. (RealsStruct_square_nonneg) We take the following as an axiom:

∀x ∈ R, zero ≤ x * x

Axiom. (RealsStruct_sum_squares_nonneg) We take the following as an axiom:

∀x y ∈ R, zero ≤ x * x + y * y

Axiom. (RealsStruct_sum_nonneg_zero_inv) We take the following as an axiom:

∀x y ∈ R, zero ≤ x → zero ≤ y → x + y = zero → x = zero ∧ y = zero

Axiom. (RealsStruct_sum_squares_zero_inv) We take the following as an axiom:

∀x y ∈ R, x * x + y * y = zero → x = zero ∧ y = zero

Axiom. (RealsStruct_leq_zero_one) We take the following as an axiom:

zero ≤ one

Primitive. The name RealsStruct_N is a term of type set.

Let N ≝ RealsStruct_N

Axiom. (RealsStruct_Arch) We take the following as an axiom:

∀x y ∈ R, zero < x → zero ≤ y → ∃n ∈ N, y ≤ n * x

Axiom. (RealsStruct_Compl) We take the following as an axiom:

∀a b ∈ R^N, (∀n ∈ N, a n ≤ b n ∧ a n ≤ a (n + one) ∧ b (n + one) ≤ b n) → ∃x ∈ R, ∀n ∈ N, a n ≤ x ∧ x ≤ b n

Axiom. (RealsStruct_natOfOrderedField) We take the following as an axiom:

explicit_Nats N zero (λm ⇒ m + one)

Definition. We define RealsStruct_Npos to be {n ∈ N|n ≠ zero}.

Let Npos ≝ RealsStruct_Npos

Axiom. (RealsStruct_PosNats_natOfOrderedField) We take the following as an axiom:

explicit_Nats Npos one (λm ⇒ m + one)

Primitive. The name RealsStruct_Z is a term of type set.

Let Z ≝ RealsStruct_Z

Primitive. The name RealsStruct_Q is a term of type set.

Let Q ≝ RealsStruct_Q

Axiom. (RealsStruct_Npos_props) We take the following as an axiom:

∀p : prop, (Npos ⊆ R → explicit_Nats Npos one (λm ⇒ m + one) → one ∈ Npos → (∀m ∈ Npos, m + one ≠ one) → (∀m ∈ Npos, ∀q : set → prop, q one → (∀n ∈ Npos, q (n + one)) → q m) → (∀n m ∈ Npos, explicit_Nats_one_plus Npos one (λm ⇒ m + one) n m = n + m) → (∀n m ∈ Npos, explicit_Nats_one_mult Npos one (λm ⇒ m + one) n m = n * m) → (∀n m ∈ Npos, n + m ∈ Npos) → (∀n m ∈ Npos, n * m ∈ Npos) → p) → p

Axiom. (RealsStruct_Npos_R) We take the following as an axiom:

Npos ⊆ R

Axiom. (RealsStruct_one_Npos) We take the following as an axiom:

one ∈ Npos

Axiom. (RealsStruct_Z_props) We take the following as an axiom:

∀p : prop, ((∀n ∈ Npos, - n ∈ Z) → zero ∈ Z → Npos ⊆ Z → Z ⊆ R → (∀n ∈ Z, ∀q : prop, (- n ∈ Npos → q) → (n = zero → q) → (n ∈ Npos → q) → q) → one ∈ Z → - one ∈ Z → (∀m ∈ Z, - m ∈ Z) → (∀n m ∈ Z, n + m ∈ Z) → (∀n m ∈ Z, n * m ∈ Z) → p) → p

Axiom. (RealsStruct_neg_Z) We take the following as an axiom:

∀n ∈ Npos, - n ∈ Z

Axiom. (RealsStruct_zero_Z) We take the following as an axiom:

zero ∈ Z

Axiom. (RealsStruct_Npos_Z) We take the following as an axiom:

Npos ⊆ Z

Axiom. (RealsStruct_Z_R) We take the following as an axiom:

Z ⊆ R

Axiom. (RealsStruct_Q_props) We take the following as an axiom:

∀p : prop, (Q ⊆ R → (∀x ∈ Q, ∀q : prop, (x ∈ R → ∀n ∈ Z, ∀m ∈ Npos, m * x = n → q) → q) → (∀x ∈ R, ∀n ∈ Z, ∀m ∈ Npos, m * x = n → x ∈ Q) → p) → p

Axiom. (RealsStruct_Q_R) We take the following as an axiom:

Q ⊆ R

Axiom. (RealsStruct_Z_Q) We take the following as an axiom:

Z ⊆ Q

Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term Field_div (Field_of_RealsStruct Rs).

Axiom. (RealsStruct_div_clos) We take the following as an axiom:

∀x ∈ R, ∀y ∈ R ∖ {zero}, x :/: y ∈ R

Axiom. (RealsStruct_mult_div) We take the following as an axiom:

∀x ∈ R, ∀y ∈ R ∖ {zero}, x = y * (x :/: y)

Axiom. (RealsStruct_div_undef1) We take the following as an axiom:

∀x y, x ∉ R → x :/: y = 0

Axiom. (RealsStruct_div_undef2) We take the following as an axiom:

∀x y, y ∉ R → x :/: y = 0

Axiom. (RealsStruct_div_undef3) We take the following as an axiom:

∀x, x :/: zero = 0

Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term CRing_with_id_omega_exp (Field_of_RealsStruct Rs).

Axiom. (RealsStruct_omega_exp_0) We take the following as an axiom:

∀x, x ^ 0 = one

Axiom. (RealsStruct_omega_exp_S) We take the following as an axiom:

∀x, ∀n ∈ ω, x ^ (ordsucc n) = x * x ^ n

Axiom. (RealsStruct_omega_exp_1) We take the following as an axiom:

∀x ∈ R, x ^ 1 = x

Axiom. (RealsStruct_omega_exp_clos) We take the following as an axiom:

∀x ∈ R, ∀n ∈ ω, x ^ n ∈ R

Primitive. The name RealsStruct_abs is a term of type set → set.

Axiom. (RealsStruct_abs_clos) We take the following as an axiom:

∀x ∈ R, RealsStruct_abs x ∈ R

Axiom. (RealsStruct_abs_nonneg_case) We take the following as an axiom: