Theorem clel5

Description: Alternate definition of class membership: a class X is an element of another class A iff there is an element of A equal to X . (Contributed by AV, 13-Nov-2020) Remove use of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 19-May-2023)

	Ref	Expression
Assertion	clel5	|- ( X e. A <-> E. x e. A X = x )

Proof

Step	Hyp	Ref	Expression
1		risset	|- ( X e. A <-> E. x e. A x = X )
2		eqcom	|- ( x = X <-> X = x )
3	2	rexbii	|- ( E. x e. A x = X <-> E. x e. A X = x )
4	1 3	bitri	|- ( X e. A <-> E. x e. A X = x )