Theorem eqv 2299

Description: The universe contains every set.

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 1473	. 2 |- (A = V <-> A.x(x e. A <-> x e. V))
2		visset 1816	. . . 4 |- x e. V
3	2	tbt 722	. . 3 |- (x e. A <-> (x e. A <-> x e. V))
4	3	albii 1001	. 2 |- (A.x x e. A <-> A.x(x e. A <-> x e. V))
5	1, 4	bitr4 176	1 |- (A = V <-> A.x x e. A)

Syntax hints:   <-> wb 146  A.wal 956   = wceq 958   e. wcel 960  Vcvv 1814

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

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