Theorem compel 16415

Description: Equivalence between two ways of saying "is a member of the complement of A."

Assertion

Ref	Expression
compel	|- (x e. (_V \ A) <-> -. x e. A)

Distinct variable group:   x,A

Proof of Theorem compel

Step	Hyp	Ref	Expression
1		eldif 2609	. 2 |- (x e. (_V \ A) <-> (x e. _V /\ -. x e. A))
2		visset 2295	. . 3 |- x e. _V
3	2	biantrur 794	. 2 |- (-. x e. A <-> (x e. _V /\ -. x e. A))
4	1, 3	bitr4i 193	1 |- (x e. (_V \ A) <-> -. x e. A)

Syntax hints:  -. wn 2   <-> wb 163   /\ wa 240   e. wcel 1300  _Vcvv 2292   \ cdif 2590

This theorem is referenced by:  compeq 16416  conss34 16419  conss1 16421

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-dif 2597

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