Theorem eqv 3434

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 2164	. 2 |- ( A = _V <-> A. x ( x e. A <-> x e. _V ) )
2		vex 2733	. . . 4 |- x e. _V
3	2	tbt 246	. . 3 |- ( x e. A <-> ( x e. A <-> x e. _V ) )
4	3	albii 1463	. 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 1346   = wceq 1348   e. wcel 2141   _Vcvv 2730

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  setindel  4522  dmi  4826