:: CARD_LAR semantic presentation begin registration cluster ($#v3_ordinal1 :::"ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#v1_card_1 :::"cardinal"::: ) -> ($#v4_ordinal1 :::"limit_ordinal"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#v3_ordinal1 :::"ordinal"::: ) ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) -> ($#v1_finset_1 :::"infinite"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#~v1_finset_1 "non" ($#v1_finset_1 :::"finite"::: ) ) ($#v1_card_1 :::"cardinal"::: ) ($#~v2_card_1 "non" ($#v2_card_1 :::"limit_cardinal"::: ) ) -> ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#v1_ordinal1 :::"epsilon-transitive"::: ) ($#v2_ordinal1 :::"epsilon-connected"::: ) ($#v3_ordinal1 :::"ordinal"::: ) ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#~v1_finset_1 "non" ($#v1_finset_1 :::"finite"::: ) ) ($#v1_card_1 :::"cardinal"::: ) ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#v1_card_5 :::"regular"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "A" be ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::); let "X" be ($#m1_hidden :::"set"::: ) ; pred "X" :::"is_unbounded_in"::: "A" means :: CARD_LAR:def 1 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) "A") & (Bool (Set ($#k3_ordinal2 :::"sup"::: ) "X") ($#r1_hidden :::"="::: ) "A") ")" ); pred "X" :::"is_closed_in"::: "A" means :: CARD_LAR:def 2 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) "A") & (Bool "(" "for" (Set (Var "B")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) "A") & (Bool (Set ($#k3_ordinal2 :::"sup"::: ) (Set "(" "X" ($#k3_xboole_0 :::"/\"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) "X") ")" ) ")" ); end; :: deftheorem defines :::"is_unbounded_in"::: CARD_LAR:def 1 : (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) ($#r1_card_lar :::"is_unbounded_in"::: ) (Set (Var "A"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Set ($#k3_ordinal2 :::"sup"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" ) ")" ))); :: deftheorem defines :::"is_closed_in"::: CARD_LAR:def 2 : (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) ($#r2_card_lar :::"is_closed_in"::: ) (Set (Var "A"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool "(" "for" (Set (Var "B")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set ($#k3_ordinal2 :::"sup"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" ) ")" ) ")" ))); definitionlet "A" be ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::); let "X" be ($#m1_hidden :::"set"::: ) ; pred "X" :::"is_club_in"::: "A" means :: CARD_LAR:def 3 (Bool "(" (Bool "X" ($#r2_card_lar :::"is_closed_in"::: ) "A") & (Bool "X" ($#r1_card_lar :::"is_unbounded_in"::: ) "A") ")" ); end; :: deftheorem defines :::"is_club_in"::: CARD_LAR:def 3 : (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) ($#r3_card_lar :::"is_club_in"::: ) (Set (Var "A"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r2_card_lar :::"is_closed_in"::: ) (Set (Var "A"))) & (Bool (Set (Var "X")) ($#r1_card_lar :::"is_unbounded_in"::: ) (Set (Var "A"))) ")" ) ")" ))); definitionlet "A" be ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "A")); attr "X" is :::"unbounded"::: means :: CARD_LAR:def 4 (Bool (Set ($#k3_ordinal2 :::"sup"::: ) "X") ($#r1_hidden :::"="::: ) "A"); attr "X" is :::"closed"::: means :: CARD_LAR:def 5 (Bool "for" (Set (Var "B")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) "A") & (Bool (Set ($#k3_ordinal2 :::"sup"::: ) (Set "(" "X" ($#k8_subset_1 :::"/\"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) "X")); end; :: deftheorem defines :::"unbounded"::: CARD_LAR:def 4 : (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"unbounded"::: ) ) "iff" (Bool (Set ($#k3_ordinal2 :::"sup"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" ))); :: deftheorem defines :::"closed"::: CARD_LAR:def 5 : (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v2_card_lar :::"closed"::: ) ) "iff" (Bool "for" (Set (Var "B")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set ($#k3_ordinal2 :::"sup"::: ) (Set "(" (Set (Var "X")) ($#k8_subset_1 :::"/\"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) ")" ))); notationlet "A" be ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "A")); antonym :::"bounded"::: "X" for :::"unbounded"::: ; end; theorem :: CARD_LAR:1 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "X")) ($#r3_card_lar :::"is_club_in"::: ) (Set (Var "A"))) "iff" (Bool "(" (Bool (Set (Var "X")) "is" ($#v2_card_lar :::"closed"::: ) ) & (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"unbounded"::: ) ) ")" ) ")" ))) ; theorem :: CARD_LAR:2 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "holds" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k3_ordinal2 :::"sup"::: ) (Set (Var "X")))))) ; theorem :: CARD_LAR:3 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Bool "not" (Set (Var "X")) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool "(" "for" (Set (Var "B1")) "being" ($#m1_hidden :::"Ordinal":::) "st" (Bool (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "ex" (Set (Var "B2")) "being" ($#m1_hidden :::"Ordinal":::) "st" (Bool "(" (Bool (Set (Var "B2")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "B2"))) ")" )) ")" )) "holds" (Bool (Set ($#k3_ordinal2 :::"sup"::: ) (Set (Var "X"))) "is" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::)))) ; theorem :: CARD_LAR:4 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"bounded"::: ) ) "iff" (Bool "ex" (Set (Var "B1")) "being" ($#m1_hidden :::"Ordinal":::) "st" (Bool "(" (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "B1"))) ")" )) ")" ))) ; theorem :: CARD_LAR:5 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Bool "not" (Set ($#k3_ordinal2 :::"sup"::: ) (Set "(" (Set (Var "X")) ($#k8_subset_1 :::"/\"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "B"))))) "holds" (Bool "ex" (Set (Var "B1")) "being" ($#m1_hidden :::"Ordinal":::) "st" (Bool "(" (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "B"))) & (Bool (Set (Set (Var "X")) ($#k8_subset_1 :::"/\"::: ) (Set (Var "B"))) ($#r1_tarski :::"c="::: ) (Set (Var "B1"))) ")" )))) ; theorem :: CARD_LAR:6 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"unbounded"::: ) ) "iff" (Bool "for" (Set (Var "B1")) "being" ($#m1_hidden :::"Ordinal":::) "st" (Bool (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "ex" (Set (Var "C")) "being" ($#m1_hidden :::"Ordinal":::) "st" (Bool "(" (Bool (Set (Var "C")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "B1")) ($#r1_ordinal1 :::"c="::: ) (Set (Var "C"))) ")" ))) ")" ))) ; theorem :: CARD_LAR:7 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"unbounded"::: ) )) "holds" (Bool "not" (Bool (Set (Var "X")) "is" ($#v1_xboole_0 :::"empty"::: ) )))) ; theorem :: CARD_LAR:8 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "B1")) "being" ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"unbounded"::: ) ) & (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "ex" (Set (Var "B3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "A")) "st" (Bool (Set (Var "B3")) ($#r2_hidden :::"in"::: ) "{" (Set (Var "B2")) where B2 "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "A")) : (Bool "(" (Bool (Set (Var "B2")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "B2"))) ")" ) "}" ))))) ; definitionlet "A" be ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "A")); let "B1" be ($#m1_hidden :::"Ordinal":::); assume (Bool (Set (Const "X")) "is" ($#v1_card_lar :::"unbounded"::: ) ) ; assume (Bool (Set (Const "B1")) ($#r2_hidden :::"in"::: ) (Set (Const "A"))) ; func :::"LBound"::: "(" "B1" "," "X" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" "X" equals :: CARD_LAR:def 6 (Set ($#k2_ordinal2 :::"inf"::: ) "{" (Set (Var "B2")) where B2 "is" ($#m1_subset_1 :::"Element"::: ) "of" "A" : (Bool "(" (Bool (Set (Var "B2")) ($#r2_hidden :::"in"::: ) "X") & (Bool "B1" ($#r2_hidden :::"in"::: ) (Set (Var "B2"))) ")" ) "}" ); end; :: deftheorem defines :::"LBound"::: CARD_LAR:def 6 : (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) (Bool "for" (Set (Var "B1")) "being" ($#m1_hidden :::"Ordinal":::) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"unbounded"::: ) ) & (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set ($#k1_card_lar :::"LBound"::: ) "(" (Set (Var "B1")) "," (Set (Var "X")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_ordinal2 :::"inf"::: ) "{" (Set (Var "B2")) where B2 "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "A")) : (Bool "(" (Bool (Set (Var "B2")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "B2"))) ")" ) "}" ))))); theorem :: CARD_LAR:9 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "B1")) "being" ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"unbounded"::: ) ) & (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "(" (Bool (Set ($#k1_card_lar :::"LBound"::: ) "(" (Set (Var "B1")) "," (Set (Var "X")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_card_lar :::"LBound"::: ) "(" (Set (Var "B1")) "," (Set (Var "X")) ")" )) ")" )))) ; theorem :: CARD_LAR:10 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) "holds" (Bool "(" (Bool (Set ($#k2_subset_1 :::"[#]"::: ) (Set (Var "A"))) "is" ($#v2_card_lar :::"closed"::: ) ) & (Bool (Set ($#k2_subset_1 :::"[#]"::: ) (Set (Var "A"))) "is" ($#v1_card_lar :::"unbounded"::: ) ) ")" )) ; theorem :: CARD_LAR:11 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "B1")) "being" ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "X")) "is" ($#v2_card_lar :::"closed"::: ) ) & (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"unbounded"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "X")) ($#k7_subset_1 :::"\"::: ) (Set (Var "B1"))) "is" ($#v2_card_lar :::"closed"::: ) ) & (Bool (Set (Set (Var "X")) ($#k7_subset_1 :::"\"::: ) (Set (Var "B1"))) "is" ($#v1_card_lar :::"unbounded"::: ) ) ")" )))) ; theorem :: CARD_LAR:12 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "B1")) "being" ($#m1_hidden :::"Ordinal":::) "st" (Bool (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "(" (Bool (Set (Set (Var "A")) ($#k6_subset_1 :::"\"::: ) (Set (Var "B1"))) "is" ($#v2_card_lar :::"closed"::: ) ) & (Bool (Set (Set (Var "A")) ($#k6_subset_1 :::"\"::: ) (Set (Var "B1"))) "is" ($#v1_card_lar :::"unbounded"::: ) ) ")" ))) ; definitionlet "A" be ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "A")); attr "X" is :::"stationary"::: means :: CARD_LAR:def 7 (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" "A" "st" (Bool (Bool (Set (Var "Y")) "is" ($#v2_card_lar :::"closed"::: ) ) & (Bool (Set (Var "Y")) "is" ($#v1_card_lar :::"unbounded"::: ) )) "holds" (Bool "not" (Bool (Set "X" ($#k9_subset_1 :::"/\"::: ) (Set (Var "Y"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))); end; :: deftheorem defines :::"stationary"::: CARD_LAR:def 7 : (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v3_card_lar :::"stationary"::: ) ) "iff" (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "Y")) "is" ($#v2_card_lar :::"closed"::: ) ) & (Bool (Set (Var "Y")) "is" ($#v1_card_lar :::"unbounded"::: ) )) "holds" (Bool "not" (Bool (Set (Set (Var "X")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Y"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))) ")" ))); theorem :: CARD_LAR:13 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v3_card_lar :::"stationary"::: ) ) & (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool (Set (Var "Y")) "is" ($#v3_card_lar :::"stationary"::: ) ))) ; definitionlet "A" be ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::); let "X" be ($#m1_hidden :::"set"::: ) ; pred "X" :::"is_stationary_in"::: "A" means :: CARD_LAR:def 8 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) "A") & (Bool "(" "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" "A" "st" (Bool (Bool (Set (Var "Y")) "is" ($#v2_card_lar :::"closed"::: ) ) & (Bool (Set (Var "Y")) "is" ($#v1_card_lar :::"unbounded"::: ) )) "holds" (Bool "not" (Bool (Set "X" ($#k9_subset_1 :::"/\"::: ) (Set (Var "Y"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) ")" ) ")" ); end; :: deftheorem defines :::"is_stationary_in"::: CARD_LAR:def 8 : (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) ($#r4_card_lar :::"is_stationary_in"::: ) (Set (Var "A"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool "(" "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "Y")) "is" ($#v2_card_lar :::"closed"::: ) ) & (Bool (Set (Var "Y")) "is" ($#v1_card_lar :::"unbounded"::: ) )) "holds" (Bool "not" (Bool (Set (Set (Var "X")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Y"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) ")" ) ")" ) ")" ))); theorem :: CARD_LAR:14 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r4_card_lar :::"is_stationary_in"::: ) (Set (Var "A"))) & (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y"))) & (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "Y")) ($#r4_card_lar :::"is_stationary_in"::: ) (Set (Var "A"))))) ; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "X")); :: original: :::"Element"::: redefine mode :::"Element"::: "of" "S" -> ($#m1_subset_1 :::"Subset":::) "of" "X"; end; theorem :: CARD_LAR:15 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v3_card_lar :::"stationary"::: ) )) "holds" (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"unbounded"::: ) ))) ; definitionlet "A" be ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "A")); func :::"limpoints"::: "X" -> ($#m1_subset_1 :::"Subset":::) "of" "A" equals :: CARD_LAR:def 9 "{" (Set (Var "B1")) where B1 "is" ($#m1_subset_1 :::"Element"::: ) "of" "A" : (Bool "(" (Bool (Set (Var "B1")) "is" ($#v1_finset_1 :::"infinite"::: ) ) & (Bool (Set (Var "B1")) "is" ($#v4_ordinal1 :::"limit_ordinal"::: ) ) & (Bool (Set ($#k3_ordinal2 :::"sup"::: ) (Set "(" "X" ($#k8_subset_1 :::"/\"::: ) (Set (Var "B1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "B1"))) ")" ) "}" ; end; :: deftheorem defines :::"limpoints"::: CARD_LAR:def 9 : (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "holds" (Bool (Set ($#k2_card_lar :::"limpoints"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "B1")) where B1 "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "A")) : (Bool "(" (Bool (Set (Var "B1")) "is" ($#v1_finset_1 :::"infinite"::: ) ) & (Bool (Set (Var "B1")) "is" ($#v4_ordinal1 :::"limit_ordinal"::: ) ) & (Bool (Set ($#k3_ordinal2 :::"sup"::: ) (Set "(" (Set (Var "X")) ($#k8_subset_1 :::"/\"::: ) (Set (Var "B1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "B1"))) ")" ) "}" ))); theorem :: CARD_LAR:16 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "B3")) "," (Set (Var "B1")) "being" ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Set (Var "X")) ($#k8_subset_1 :::"/\"::: ) (Set (Var "B3"))) ($#r1_tarski :::"c="::: ) (Set (Var "B1")))) "holds" (Bool (Set (Set (Var "B3")) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_card_lar :::"limpoints"::: ) (Set (Var "X")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_ordinal1 :::"succ"::: ) (Set (Var "B1"))))))) ; theorem :: CARD_LAR:17 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "B1")) "being" ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "B1")))) "holds" (Bool (Set ($#k2_card_lar :::"limpoints"::: ) (Set (Var "X"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_ordinal1 :::"succ"::: ) (Set (Var "B1"))))))) ; theorem :: CARD_LAR:18 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "holds" (Bool (Set ($#k2_card_lar :::"limpoints"::: ) (Set (Var "X"))) "is" ($#v2_card_lar :::"closed"::: ) ))) ; theorem :: CARD_LAR:19 (Bool "for" (Set (Var "A")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"unbounded"::: ) ) & (Bool (Set ($#k2_card_lar :::"limpoints"::: ) (Set (Var "X"))) "is" ($#v1_card_lar :::"bounded"::: ) )) "holds" (Bool "ex" (Set (Var "B1")) "being" ($#m1_hidden :::"Ordinal":::) "st" (Bool "(" (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool "{" (Set "(" ($#k1_ordinal1 :::"succ"::: ) (Set (Var "B2")) ")" ) where B2 "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "A")) : (Bool "(" (Bool (Set (Var "B2")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_ordinal1 :::"succ"::: ) (Set (Var "B2")))) ")" ) "}" ($#r3_card_lar :::"is_club_in"::: ) (Set (Var "A"))) ")" )))) ; registrationlet "M" be ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::); cluster ($#v1_ordinal1 :::"epsilon-transitive"::: ) ($#v2_ordinal1 :::"epsilon-connected"::: ) ($#v3_ordinal1 :::"ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#v1_card_1 :::"cardinal"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" "M"; end; theorem :: CARD_LAR:20 (Bool "for" (Set (Var "M")) "being" ($#m1_hidden :::"Aleph":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "M")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"unbounded"::: ) )) "holds" (Bool (Set ($#k1_card_5 :::"cf"::: ) (Set (Var "M"))) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X")))))) ; theorem :: CARD_LAR:21 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "M")) "st" (Bool (Bool "(" "for" (Set (Var "X")) "being" ($#m1_card_lar :::"Element"::: ) "of" (Set (Var "S")) "holds" (Bool (Set (Var "X")) "is" ($#v2_card_lar :::"closed"::: ) ) ")" )) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "S"))) "is" ($#v2_card_lar :::"closed"::: ) ))) ; theorem :: CARD_LAR:22 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "M")) "st" (Bool (Bool (Set ($#k4_ordinal1 :::"omega"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_card_5 :::"cf"::: ) (Set (Var "M"))))) "holds" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set (Var "X")) "holds" (Bool (Set ($#k3_ordinal2 :::"sup"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "M")))))) ; theorem :: CARD_LAR:23 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) "st" (Bool (Bool (Set ($#k4_ordinal1 :::"omega"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_card_5 :::"cf"::: ) (Set (Var "M"))))) "holds" (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "M")) "st" (Bool (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_card_5 :::"cf"::: ) (Set (Var "M")))) & (Bool "(" "for" (Set (Var "X")) "being" ($#m1_card_lar :::"Element"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v2_card_lar :::"closed"::: ) ) & (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"unbounded"::: ) ) ")" ) ")" )) "holds" (Bool "(" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "S"))) "is" ($#v2_card_lar :::"closed"::: ) ) & (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "S"))) "is" ($#v1_card_lar :::"unbounded"::: ) ) ")" ))) ; theorem :: CARD_LAR:24 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "M")) "st" (Bool (Bool (Set ($#k4_ordinal1 :::"omega"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_card_5 :::"cf"::: ) (Set (Var "M")))) & (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"unbounded"::: ) )) "holds" (Bool "for" (Set (Var "B1")) "being" ($#m1_hidden :::"Ordinal":::) "st" (Bool (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "M")))) "holds" (Bool "ex" (Set (Var "B")) "being" ($#v4_ordinal1 :::"limit_ordinal"::: ) ($#v1_finset_1 :::"infinite"::: ) ($#m1_hidden :::"Ordinal":::) "st" (Bool "(" (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "M"))) & (Bool (Set (Var "B1")) ($#r2_hidden :::"in"::: ) (Set (Var "B"))) & (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set ($#k2_card_lar :::"limpoints"::: ) (Set (Var "X")))) ")" ))))) ; theorem :: CARD_LAR:25 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "M")) "st" (Bool (Bool (Set ($#k4_ordinal1 :::"omega"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_card_5 :::"cf"::: ) (Set (Var "M")))) & (Bool (Set (Var "X")) "is" ($#v1_card_lar :::"unbounded"::: ) )) "holds" (Bool (Set ($#k2_card_lar :::"limpoints"::: ) (Set (Var "X"))) "is" ($#v1_card_lar :::"unbounded"::: ) ))) ; definitionlet "M" be ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::); attr "M" is :::"Mahlo"::: means :: CARD_LAR:def 10 (Bool "{" (Set (Var "N")) where N "is" ($#v1_finset_1 :::"infinite"::: ) ($#v1_card_1 :::"cardinal"::: ) ($#m1_subset_1 :::"Element"::: ) "of" "M" : (Bool (Set (Var "N")) "is" ($#v1_card_5 :::"regular"::: ) ) "}" ($#r4_card_lar :::"is_stationary_in"::: ) "M"); attr "M" is :::"strongly_Mahlo"::: means :: CARD_LAR:def 11 (Bool "{" (Set (Var "N")) where N "is" ($#v1_finset_1 :::"infinite"::: ) ($#v1_card_1 :::"cardinal"::: ) ($#m1_subset_1 :::"Element"::: ) "of" "M" : (Bool (Set (Var "N")) "is" ($#v6_card_fil :::"strongly_inaccessible"::: ) ) "}" ($#r4_card_lar :::"is_stationary_in"::: ) "M"); end; :: deftheorem defines :::"Mahlo"::: CARD_LAR:def 10 : (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v4_card_lar :::"Mahlo"::: ) ) "iff" (Bool "{" (Set (Var "N")) where N "is" ($#v1_finset_1 :::"infinite"::: ) ($#v1_card_1 :::"cardinal"::: ) ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "M")) : (Bool (Set (Var "N")) "is" ($#v1_card_5 :::"regular"::: ) ) "}" ($#r4_card_lar :::"is_stationary_in"::: ) (Set (Var "M"))) ")" )); :: deftheorem defines :::"strongly_Mahlo"::: CARD_LAR:def 11 : (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v5_card_lar :::"strongly_Mahlo"::: ) ) "iff" (Bool "{" (Set (Var "N")) where N "is" ($#v1_finset_1 :::"infinite"::: ) ($#v1_card_1 :::"cardinal"::: ) ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "M")) : (Bool (Set (Var "N")) "is" ($#v6_card_fil :::"strongly_inaccessible"::: ) ) "}" ($#r4_card_lar :::"is_stationary_in"::: ) (Set (Var "M"))) ")" )); theorem :: CARD_LAR:26 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v5_card_lar :::"strongly_Mahlo"::: ) )) "holds" (Bool (Set (Var "M")) "is" ($#v4_card_lar :::"Mahlo"::: ) )) ; theorem :: CARD_LAR:27 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v4_card_lar :::"Mahlo"::: ) )) "holds" (Bool (Set (Var "M")) "is" ($#v1_card_5 :::"regular"::: ) )) ; theorem :: CARD_LAR:28 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v4_card_lar :::"Mahlo"::: ) )) "holds" (Bool (Set (Var "M")) "is" ($#v2_card_1 :::"limit_cardinal"::: ) )) ; theorem :: CARD_LAR:29 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v4_card_lar :::"Mahlo"::: ) )) "holds" (Bool (Set (Var "M")) "is" ($#v4_card_fil :::"inaccessible"::: ) )) ; theorem :: CARD_LAR:30 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v5_card_lar :::"strongly_Mahlo"::: ) )) "holds" (Bool (Set (Var "M")) "is" ($#v5_card_fil :::"strong_limit"::: ) )) ; theorem :: CARD_LAR:31 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v5_card_lar :::"strongly_Mahlo"::: ) )) "holds" (Bool (Set (Var "M")) "is" ($#v6_card_fil :::"strongly_inaccessible"::: ) )) ; begin theorem :: CARD_LAR:32 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set (Var "y"))) & (Bool (Set (Var "y")) "is" ($#m1_hidden :::"Cardinal":::)) ")" )) ")" )) "holds" (Bool (Set ($#k3_tarski :::"union"::: ) (Set (Var "X"))) "is" ($#m1_hidden :::"Cardinal":::))) ; theorem :: CARD_LAR:33 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m1_hidden :::"Aleph":::) "st" (Bool (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_card_5 :::"cf"::: ) (Set (Var "M")))) & (Bool "(" "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "Y")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "Y"))) ($#r2_hidden :::"in"::: ) (Set (Var "M"))) ")" )) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "X")) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "M"))))) ; theorem :: CARD_LAR:34 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"Ordinal":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v6_card_fil :::"strongly_inaccessible"::: ) ) & (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "M")))) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k4_classes1 :::"Rank"::: ) (Set (Var "A")) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "M"))))) ; theorem :: CARD_LAR:35 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v6_card_fil :::"strongly_inaccessible"::: ) )) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k4_classes1 :::"Rank"::: ) (Set (Var "M")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "M")))) ; theorem :: CARD_LAR:36 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v6_card_fil :::"strongly_inaccessible"::: ) )) "holds" (Bool (Set ($#k4_classes1 :::"Rank"::: ) (Set (Var "M"))) "is" ($#v2_classes1 :::"Tarski"::: ) )) ; theorem :: CARD_LAR:37 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"Ordinal":::) "holds" (Bool (Bool "not" (Set ($#k4_classes1 :::"Rank"::: ) (Set (Var "A"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))) ; registrationlet "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"Ordinal":::); cluster (Set ($#k4_classes1 :::"Rank"::: ) "A") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: CARD_LAR:38 (Bool "for" (Set (Var "M")) "being" ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ($#m1_hidden :::"Aleph":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v6_card_fil :::"strongly_inaccessible"::: ) )) "holds" (Bool (Set ($#k4_classes1 :::"Rank"::: ) (Set (Var "M"))) "is" ($#m1_hidden :::"Universe":::))) ;