Theorem elisset

Description: An element of a class exists. (Contributed by NM, 1-May-1995) Reduce dependencies on axioms. (Revised by BJ, 29-Apr-2019)

	Ref	Expression
Assertion	elisset	|- ( A e. V -> E. x x = A )

Proof

Step	Hyp	Ref	Expression
1		elissetv	|- ( A e. V -> E. y y = A )
2		vextru	|- y e. { z | T. }
3	2	biantru	|- ( y = A <-> ( y = A /\ y e. { z | T. } ) )
4	3	exbii	|- ( E. y y = A <-> E. y ( y = A /\ y e. { z | T. } ) )
5		dfclel	|- ( A e. { z | T. } <-> E. y ( y = A /\ y e. { z | T. } ) )
6	4 5	bitr4i	|- ( E. y y = A <-> A e. { z | T. } )
7		vextru	|- x e. { z | T. }
8	7	biantru	|- ( x = A <-> ( x = A /\ x e. { z | T. } ) )
9	8	exbii	|- ( E. x x = A <-> E. x ( x = A /\ x e. { z | T. } ) )
10		dfclel	|- ( A e. { z | T. } <-> E. x ( x = A /\ x e. { z | T. } ) )
11	9 10	bitr4i	|- ( E. x x = A <-> A e. { z | T. } )
12	6 11	bitr4i	|- ( E. y y = A <-> E. x x = A )
13	1 12	sylib	|- ( A e. V -> E. x x = A )