Theorem eqv 3321

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 2089	. 2 |- ( A = _V <-> A. x ( x e. A <-> x e. _V ) )
2		vex 2636	. . . 4 |- x e. _V
3	2	tbt 246	. . 3 |- ( x e. A <-> ( x e. A <-> x e. _V ) )
4	3	albii 1411	. 2 |- ( A. x x e. A <-> A. x ( x e. A <-> x e. _V ) )
5	1, 4	bitr4i 186	1 |- ( A = _V <-> A. x x e. A )

Syntax hints:   <-> wb 104   A.wal 1294   = wceq 1296   e. wcel 1445   _Vcvv 2633

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-v 2635

This theorem is referenced by:  setindel  4382  dmi  4682