Theorem zfcndun

Description: Axiom of Union ax-un , reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 15-Aug-2003) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	zfcndun	$$⊢\exists y\forall z\left(\exists w\left(z\in w\wedge w\in x\right)\to z\in y\right)$$

Proof

Step	Hyp	Ref	Expression
1		axunnd	$$⊢\exists y\forall z\left(\exists y\left(z\in y\wedge y\in x\right)\to z\in y\right)$$
2		elequ2	$$⊢w=y\to \left(z\in w\leftrightarrow z\in y\right)$$
3		elequ1	$$⊢w=y\to \left(w\in x\leftrightarrow y\in x\right)$$
4	2 3	anbi12d	$$⊢w=y\to \left(\left(z\in w\wedge w\in x\right)\leftrightarrow \left(z\in y\wedge y\in x\right)\right)$$
5	4	cbvexvw	$$⊢\exists w\left(z\in w\wedge w\in x\right)\leftrightarrow \exists y\left(z\in y\wedge y\in x\right)$$
6	5	imbi1i	$$⊢\left(\exists w\left(z\in w\wedge w\in x\right)\to z\in y\right)\leftrightarrow \left(\exists y\left(z\in y\wedge y\in x\right)\to z\in y\right)$$
7	6	albii	$$⊢\forall z\left(\exists w\left(z\in w\wedge w\in x\right)\to z\in y\right)\leftrightarrow \forall z\left(\exists y\left(z\in y\wedge y\in x\right)\to z\in y\right)$$
8	7	exbii	$$⊢\exists y\forall z\left(\exists w\left(z\in w\wedge w\in x\right)\to z\in y\right)\leftrightarrow \exists y\forall z\left(\exists y\left(z\in y\wedge y\in x\right)\to z\in y\right)$$
9	1 8	mpbir	$$⊢\exists y\forall z\left(\exists w\left(z\in w\wedge w\in x\right)\to z\in y\right)$$