\documentclass{article}
\newtheorem{axiom}{Axiom}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
\newcommand\unicdcdcdbg{\cup}
\newcommand\unicdcdbdgd{\setminus}
\newcommand\unicdcdcdjd{\cap}
\newcommand\unicdcdadad{\forall}
\newcommand\unicdbdjdcd{\Rightarrow}
\newcommand\unicdcdadjd{\notin}
\newcommand\unicdbdjded{\Leftrightarrow}
\newcommand\unicdcdcdid{\lor}
\newcommand\unicdcdcdhd{\land}
\newcommand\uniddcgcg{\lambda}
\newcommand\unicdcdaddd{\exists}
\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{thm\_2Eenumeral\_2ENOT\_\_IN\_\_nt (TMdeTmkDRZ9DvnM2ewhw4jTgrMC1QGcAUvi)}
\maketitle
Let $ ty\_2EternaryComparisons\_2Eordering : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2EternaryComparisons\_2Eordering}}\label{hyp:ogfgpfehjhpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfepgchegfgchjgoghg}
\end{equation}
Let $ c\_2EternaryComparisons\_2ELESS : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2EternaryComparisons\_2ELESS} \in {ty\_2EternaryComparisons\_2Eordering}}\label{hyp:ngfgngpfdgpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfemefedfdf}
\end{equation}
Let $ ty\_2Etoto\_2Etoto : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Etoto\_2Etoto}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfeehpgehpgpfcdfeehpgehpg}
\end{equation}
Let $ c\_2Etoto\_2Eapto : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Etoto\_2Eapto~A\_27a \in (((ty\_2EternaryComparisons\_2Eordering^{{A\_27a}})^{{A\_27a}})^{{({{ty\_2Etoto\_2Etoto}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeehpgehpgpfcdfebgahehpg}
\end{equation}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepffddepfcdge}
We define $\mname{c\_2Ebool\_2E\_5C\_2F}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}}~{V2t}})}})}})}})}})}})}$.
\end{definition}
Let $ ty\_2Eenumeral\_2Ebt : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Eenumeral\_2Ebt}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfefgogfhngfgchbgmgpfcdfecgeh}
\end{equation}
Let $ c\_2Eenumeral\_2Enode : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Eenumeral\_2Enode~A\_27a \in (((\\
(ty\_2Eenumeral\_2Ebt~A\_27a)^{{({{ty\_2Eenumeral\_2Ebt}~{A\_27a}})}})^{{A\_27a}})^{{({{ty\_2Eenumeral\_2Ebt}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfefgogfhngfgchbgmgpfcdfeogpgegfg}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
Let $ c\_2Eenumeral\_2Ent : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Eenumeral\_2Ent~A\_27a \in (ty\_2Eenumeral\_2Ebt~\\
A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfefgogfhngfgchbgmgpfcdfeogeh}
\end{equation}
Let $ c\_2Eenumeral\_2EENUMERAL : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Eenumeral\_2EENUMERAL~A\_27a \in \\
(((2^{{A\_27a}})^{{({{ty\_2Eenumeral\_2Ebt}~{A\_27a}})}})^{{({{ty\_2Etoto\_2Etoto}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfefgogfhngfgchbgmgpfcdfefeoeffnefecfbeme}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfejeoe}
We define $\mname{c\_2Ebool\_2EIN}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1f \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V1f}}~{V0x}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdgepffdde}
We define $\mname{c\_2Ebool\_2E\_2F\_5C}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}})}}~{V2t}})}})}})}})}})}$.
\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0cmp \in (ty\_2Etoto\_2Etoto~\\
A\_27a) .  ({\unicdcdadad} V1y \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V1y)~\\
(ap~(ap~(c\_2Eenumeral\_2EENUMERAL~A\_27a)~V0cmp)~(c\_2Eenumeral\_2Ent~\\
A\_27a)))){\unicdbdjded}False))){\unicdcdcdhd}({\unicdcdadad} V2cmp \in (ty\_2Etoto\_2Etoto~A\_27a) .  \\
({\unicdcdadad} V3l \in (ty\_2Eenumeral\_2Ebt~A\_27a) .  ({\unicdcdadad} V4x \in A\_27a .  \\
({\unicdcdadad} V5r \in (ty\_2Eenumeral\_2Ebt~A\_27a) .  ({\unicdcdadad} V6y \in A\_27a .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V6y)~(ap~(ap~(c\_2Eenumeral\_2EENUMERAL~\\
A\_27a)~V2cmp)~(ap~(ap~(ap~(c\_2Eenumeral\_2Enode~A\_27a)~V3l)~V4x)~\\
V5r)))){\unicdbdjded}(((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V6y)~(ap~(ap~(c\_2Eenumeral\_2EENUMERAL~\\
A\_27a)~V2cmp)~V3l))){\unicdcdcdhd}((ap~(ap~(ap~(c\_2Etoto\_2Eapto~A\_27a)~V2cmp)~\\
V6y)~V4x)=c\_2EternaryComparisons\_2ELESS)){\unicdcdcdid}((V6y=V4x){\unicdcdcdid}((p~\\
(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V6y)~(ap~(ap~(c\_2Eenumeral\_2EENUMERAL~\\
A\_27a)~V2cmp)~V5r))){\unicdcdcdhd}((ap~(ap~(ap~(c\_2Etoto\_2Eapto~A\_27a)~V2cmp)~\\
V4x)~V6y)=c\_2EternaryComparisons\_2ELESS)))))))))))\end{array}
\label{hyp:ehigngpfcdfefgogfhngfgchbgmgpfcdfejeoepfpfcgehpfpfehpgpfpfdhfgeh}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfefgogfhngfgchbgmgpfcdfeoepeefpfpfjeoepfpfogeh}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0cmp \in (ty\_2Etoto\_2Etoto~\\
A\_27a) .  ({\unicdcdadad} V1y \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V1y)~\\
(ap~(ap~(c\_2Eenumeral\_2EENUMERAL~A\_27a)~V0cmp)~(c\_2Eenumeral\_2Ent~\\
A\_27a)))){\unicdbdjded}False)))\end{array}$$
\end{theorem}
\end{document}
