Theorem undif 2339

Description: Union of complementary parts into whole.

Assertion

Ref	Expression
undif	|- (A (_ B <-> (A u. (B \ A)) = B)

Proof of Theorem undif

Step	Hyp	Ref	Expression
1		ssequn1 2196	. 2 |- (A (_ B <-> (A u. B) = B)
2		undif2 2337	. . 3 |- (A u. (B \ A)) = (A u. B)
3	2	eqeq1i 1479	. 2 |- ((A u. (B \ A)) = B <-> (A u. B) = B)
4	1, 3	bitr4 176	1 |- (A (_ B <-> (A u. (B \ A)) = B)

Syntax hints:   <-> wb 146   = wceq 954   \ cdif 2040   u. cun 2041   (_ wss 2043

This theorem is referenced by:  difsnid 2463  dfdom2 4371  sbthlem5 4437  sbthlem6 4438  fodomr 4469  mapdom2 4480  limensuci 4492  unfi 4534  xrsupss 6033  xrinfmss 6034

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277

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