a-3
\(\bigcap.\)【\(x \in \bigcap X \Longleftrightarrow \begin{cases} \exists x \, x \in X \Rightarrow \forall y \in X . x \in y \\ \not\exists x \, x \in X \Rightarrow x \in \bigcap X \end{cases}\)】\(\,{\triangleleft}\,\)b-1
\(\text{ax_r}\)【\(\forall X ( \exists x \, x \in X \Rightarrow \exists y \in X . \not\exists x \, x \in X , x \in y )\)】\(\,{\triangleleft}\,\)\({0.}\)【\(x \in 0 \Longleftrightarrow {\perp}\)】\(\,{\triangleleft}\,\)
\({\tt n}.\)【\(x \in {\tt n} \Longleftrightarrow x = 0 \mathbin{\rm o\!r} x = 1 \mathbin{\rm o\!r} \cdots \mathbin{\rm o\!r} x = {\tt n\,{\text -}\,1}\)】\(\,{\triangleleft}\,\)
def …【\(\text{Ind}\)】≃【\(\{ M \mid 0 \in M , \forall m \in M . m +1 \in M \}\)】
【\(\text{Ind} = \text{Ind}\)】\(\,{\blacktriangleleft}\,\mathbb{W}.\)
\(\mathbb{M}1\)【\(\forall n \in \mathbb{M} . n +1 \in \mathbb{M}\)】\(\,{\blacktriangleleft}\,\mathbb{W}.\)
\({+1}.\)【\(x \in m +1 \Longleftrightarrow \begin{cases} m \in \mathbb{M} \Rightarrow ( x \in m \mathbin{\rm o\!r} x = m ) \\ m \notin \mathbb{M} \Rightarrow x \in m +1 \end{cases}\)】\(\,{\triangleleft}\,\)
\(\mathbb{N}.\)【\(n \in \mathbb{N} \Longleftrightarrow 0 \neq n \in \mathbb{M}\)】\(\,{\triangleleft}\,\)
c-2
\({\stackrel{\rm I}\to}.\)【\(X \stackrel{\rm I}\to Y = \{ f \in X \to Y \mid \forall y \, ! x \, \langle x , y \rangle \in f \}\)】\({\stackrel{\rm S}\to}.\)【\(X \stackrel{\rm S}\to Y = \{ f \in X \to Y \mid \forall y \in Y . \exists x \in X . \langle x , y \rangle \in f \}\)】
\(\stackrel{\#}=.\)【\(X \stackrel{\#}= Y \Longleftrightarrow \exists f \, f \in X \stackrel{\rm IS}\to Y\)】
\(\stackrel{\#}\le.\)【\(X \stackrel{\#}\le Y \Longleftrightarrow \exists f \, f \in X \stackrel{\rm I}\to Y\)】