Theorem clel5 2909

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.) (Revised by Steven Nguyen, 19-May-2023.)

Assertion

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

Distinct variable groups:   x, A   x, X

Proof of Theorem clel5

Step	Hyp	Ref	Expression
1		risset 2533	. 2 |- ( X e. A <-> E. x e. A x = X )
2		eqcom 2206	. . 3 |- ( x = X <-> X = x )
3	2	rexbii 2512	. 2 |- ( E. x e. A x = X <-> E. x e. A X = x )
4	1, 3	bitri 184	1 |- ( X e. A <-> E. x e. A X = x )

Syntax hints:   <-> wb 105   = wceq 1372   e. wcel 2175   E.wrex 2484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-cleq 2197  df-clel 2200  df-rex 2489

This theorem is referenced by:  wrdlen1  11029  phisum  12534