Theorem eqv 3470

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 2190	. 2 |- ( A = _V <-> A. x ( x e. A <-> x e. _V ) )
2		vex 2766	. . . 4 |- x e. _V
3	2	tbt 247	. . 3 |- ( x e. A <-> ( x e. A <-> x e. _V ) )
4	3	albii 1484	. 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 1362   = wceq 1364   e. wcel 2167   _Vcvv 2763

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765

This theorem is referenced by:  setindel  4574  dmi  4881