Theorem eqv 3440

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 2169	. 2 |- ( A = _V <-> A. x ( x e. A <-> x e. _V ) )
2		vex 2738	. . . 4 |- x e. _V
3	2	tbt 247	. . 3 |- ( x e. A <-> ( x e. A <-> x e. _V ) )
4	3	albii 1468	. 2 |- ( A. x x e. A <-> A. x ( x e. A <-> x e. _V ) )
5	1, 4	bitr4i 187	1 |- ( A = _V <-> A. x x e. A )

Syntax hints:   <-> wb 105   A.wal 1351   = wceq 1353   e. wcel 2146   _Vcvv 2735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-v 2737

This theorem is referenced by:  setindel  4531  dmi  4835