Theorem eqv 3566

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 2347	. 2 |- (A = _V <-> A.x(x e. A <-> x e. _V))
2		vex 2863	. . . 4 |- x e. _V
3	2	tbt 333	. . 3 |- (x e. A <-> (x e. A <-> x e. _V))
4	3	albii 1566	. 2 |- (A.x x e. A <-> A.x(x e. A <-> x e. _V))
5	1, 4	bitr4i 243	1 |- (A = _V <-> A.x x e. A)

Syntax hints:   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  nincompl  4073  xpvv  4844  dmi  4920  1stfo  5506  2ndfo  5507  swapf1o  5512  dmep  5525