Theorem wfaxun

Description: The class of well-founded sets models the Axiom of Union ax-un . Part of Corollary II.2.5 of Kunen2 p. 112. (Contributed by Eric Schmidt, 19-Oct-2025)

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

Proof

Step	Hyp	Ref	Expression
1		wfax.1	$$⊢W=\bigcup R_1\left[\mathrm{On}\right]$$
2		uniclaxun	$$⊢\forall x\in W\bigcup x\in W\to \forall x\in W\exists y\in W\forall z\in W\left(\exists w\in W\left(z\in w\wedge w\in x\right)\to z\in y\right)$$
3		uniwf	$$⊢x\in \bigcup R_1\left[\mathrm{On}\right]\leftrightarrow \bigcup x\in \bigcup R_1\left[\mathrm{On}\right]$$
4	1	eleq2i	$$⊢x\in W\leftrightarrow x\in \bigcup R_1\left[\mathrm{On}\right]$$
5	1	eleq2i	$$⊢\bigcup x\in W\leftrightarrow \bigcup x\in \bigcup R_1\left[\mathrm{On}\right]$$
6	3 4 5	3bitr4i	$$⊢x\in W\leftrightarrow \bigcup x\in W$$
7	6	biimpi	$$⊢x\in W\to \bigcup x\in W$$
8	2 7	mprg	$$⊢\forall x\in W\exists y\in W\forall z\in W\left(\exists w\in W\left(z\in w\wedge w\in x\right)\to z\in y\right)$$