Theorem eqv 2890

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 1878	. 2 |- (A = _V <-> A.x(x e. A <-> x e. _V))
2		visset 2295	. . . 4 |- x e. _V
3	2	tbt 788	. . 3 |- (x e. A <-> (x e. A <-> x e. _V))
4	3	albii 1346	. 2 |- (A.x x e. A <-> A.x(x e. A <-> x e. _V))
5	1, 4	bitr4i 193	1 |- (A = _V <-> A.x x e. A)

Syntax hints:   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  _Vcvv 2292

This theorem is referenced by:  dmi 4172  fnbigcup 14075  dominc 14622  rninc 14623

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

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