Theorem elisset

Description: An element of a class exists. Use elissetv instead when sufficient (for instance in usages where x is a dummy variable). (Contributed by NM, 1-May-1995) Reduce dependencies on axioms. (Revised by BJ, 29-Apr-2019)

	Ref	Expression
Assertion	elisset	$$⊢A\in V\to \exists xx=A$$

Proof

Step	Hyp	Ref	Expression
1		elissetv	$$⊢A\in V\to \exists yy=A$$
2		vextru	$$⊢y\in \left\{z|\top \right\}$$
3	2	issetlem	$$⊢A\in \left\{z|\top \right\}\leftrightarrow \exists yy=A$$
4		vextru	$$⊢x\in \left\{z|\top \right\}$$
5	4	issetlem	$$⊢A\in \left\{z|\top \right\}\leftrightarrow \exists xx=A$$
6	3 5	bitr3i	$$⊢\exists yy=A\leftrightarrow \exists xx=A$$
7	1 6	sylib	$$⊢A\in V\to \exists xx=A$$