Theorem clel5 2849

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 2485	. 2 |- ( X e. A <-> E. x e. A x = X )
2		eqcom 2159	. . 3 |- ( x = X <-> X = x )
3	2	rexbii 2464	. 2 |- ( E. x e. A x = X <-> E. x e. A X = x )
4	1, 3	bitri 183	1 |- ( X e. A <-> E. x e. A X = x )

Syntax hints:   <-> wb 104   = wceq 1335   e. wcel 2128   E.wrex 2436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-cleq 2150  df-clel 2153  df-rex 2441

This theorem is referenced by:  phisum  12103