Theorem eqv 3240

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 2034	. 2 |- ( A = _V <-> A. x ( x e. A <-> x e. _V ) )
2		vex 2560	. . . 4 |- x e. _V
3	2	tbt 236	. . 3 |- ( x e. A <-> ( x e. A <-> x e. _V ) )
4	3	albii 1359	. 2 |- ( A. x x e. A <-> A. x ( x e. A <-> x e. _V ) )
5	1, 4	bitr4i 176	1 |- ( A = _V <-> A. x x e. A )

Syntax hints:   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  setindel  4263  dmi  4550