Theorem axexte

Description: The axiom of extensionality ( ax-ext ) restated so that it postulates the existence of a set z given two arbitrary sets x and y . This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets. (Contributed by NM, 28-Sep-2003)

	Ref	Expression
Assertion	axexte	$$⊢\exists z\left(\left(z\in x\leftrightarrow z\in y\right)\to x=y\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-ext	$$⊢\forall z\left(z\in x\leftrightarrow z\in y\right)\to x=y$$
2		19.36v	$$⊢\exists z\left(\left(z\in x\leftrightarrow z\in y\right)\to x=y\right)\leftrightarrow \left(\forall z\left(z\in x\leftrightarrow z\in y\right)\to x=y\right)$$
3	1 2	mpbir	$$⊢\exists z\left(\left(z\in x\leftrightarrow z\in y\right)\to x=y\right)$$