Theorem undif 2343

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 2200	. 2 |- (A (_ B <-> (A u. B) = B)
2		undif2 2341	. . 3 |- (A u. (B \ A)) = (A u. B)
3	2	eqeq1i 1482	. 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 956   \ cdif 2044   u. cun 2045   (_ wss 2047

This theorem is referenced by:  difsnid 2467  dfdom2 4384  sbthlem5 4451  sbthlem6 4452  fodomr 4483  mapdom2 4494  limensuci 4506  unfi 4551  unfiOLD 4552  xrsupss 6078  xrinfmss 6079  rcfpfillem6 10595  rcfpfillem6OLD 10596

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281

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