Theorem clel5 2940

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 2558	. 2 |- ( X e. A <-> E. x e. A x = X )
2		eqcom 2231	. . 3 |- ( x = X <-> X = x )
3	2	rexbii 2537	. 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 1395   e. wcel 2200   E.wrex 2509

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-cleq 2222  df-clel 2225  df-rex 2514

This theorem is referenced by:  wrdlen1  11104  phisum  12758