Theorem clel5 3343

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.)

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		id 22	. . 3 |- ( X e. A -> X e. A )
2		eqeq2 2633	. . . 4 |- ( x = X -> ( X = x <-> X = X ) )
3	2	adantl 482	. . 3 |- ( ( X e. A /\ x = X ) -> ( X = x <-> X = X ) )
4		eqidd 2623	. . 3 |- ( X e. A -> X = X )
5	1, 3, 4	rspcedvd 3317	. 2 |- ( X e. A -> E. x e. A X = x )
6		eleq1a 2696	. . 3 |- ( x e. A -> ( X = x -> X e. A ) )
7	6	rexlimiv 3027	. 2 |- ( E. x e. A X = x -> X e. A )
8	5, 7	impbii 199	1 |- ( X e. A <-> E. x e. A X = x )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by:  dfss5  3864  disjunsn  29407