Theorem eqv 3267

Description: The universe contains every set. (Contributed by NM, 11-Sep-2006.)

Assertion

Ref	Expression
eqv	|- ( A = _V <-> A. x x e. A )

Distinct variable group:   x, A

Proof of Theorem eqv

Step	Hyp	Ref	Expression
1		dfcleq 2075	. 2 |- ( A = _V <-> A. x ( x e. A <-> x e. _V ) )
2		vex 2604	. . . 4 |- x e. _V
3	2	tbt 245	. . 3 |- ( x e. A <-> ( x e. A <-> x e. _V ) )
4	3	albii 1399	. 2 |- ( A. x x e. A <-> A. x ( x e. A <-> x e. _V ) )
5	1, 4	bitr4i 185	1 |- ( A = _V <-> A. x x e. A )

Syntax hints:   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   _Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by:  setindel  4281  dmi  4568