Theorem wfaxinf2

Description: The class of well-founded sets models the Axiom of Infinity ax-inf2 . Part of Corollary II.2.12 of Kunen2 p. 114. (Contributed by Eric Schmidt, 19-Oct-2025)

	Ref	Expression
Hypothesis	wfax.1	$$⊢W=\bigcup R_1\left[\mathrm{On}\right]$$
Assertion	wfaxinf2	$$⊢\exists x\in W\left(\exists y\in W\left(y\in x\wedge \forall z\in W\neg z\in y\right)\wedge \forall y\in W\left(y\in x\to \exists z\in W\left(z\in x\wedge \forall w\in W\left(w\in z\leftrightarrow \left(w\in y\vee w=y\right)\right)\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		wfax.1	$$⊢W=\bigcup R_1\left[\mathrm{On}\right]$$
2		trwf	$$⊢\mathrm{Tr}\bigcup R_1\left[\mathrm{On}\right]$$
3		treq	$$⊢W=\bigcup R_1\left[\mathrm{On}\right]\to \left(\mathrm{Tr}W\leftrightarrow \mathrm{Tr}\bigcup R_1\left[\mathrm{On}\right]\right)$$
4	1 3	ax-mp	$$⊢\mathrm{Tr}W\leftrightarrow \mathrm{Tr}\bigcup R_1\left[\mathrm{On}\right]$$
5	2 4	mpbir	$$⊢\mathrm{Tr}W$$
6		onwf	$$⊢\mathrm{On}\subseteq \bigcup R_1\left[\mathrm{On}\right]$$
7		omelon	$$⊢\mathrm{\omega }\in \mathrm{On}$$
8	6 7	sselii	$$⊢\mathrm{\omega }\in \bigcup R_1\left[\mathrm{On}\right]$$
9	8 1	eleqtrri	$$⊢\mathrm{\omega }\in W$$
10		omelaxinf2	$$⊢\left(\mathrm{Tr}W\wedge \mathrm{\omega }\in W\right)\to \exists x\in W\left(\exists y\in W\left(y\in x\wedge \forall z\in W\neg z\in y\right)\wedge \forall y\in W\left(y\in x\to \exists z\in W\left(z\in x\wedge \forall w\in W\left(w\in z\leftrightarrow \left(w\in y\vee w=y\right)\right)\right)\right)\right)$$
11	5 9 10	mp2an	$$⊢\exists x\in W\left(\exists y\in W\left(y\in x\wedge \forall z\in W\neg z\in y\right)\wedge \forall y\in W\left(y\in x\to \exists z\in W\left(z\in x\wedge \forall w\in W\left(w\in z\leftrightarrow \left(w\in y\vee w=y\right)\right)\right)\right)\right)$$