Theorem undif2 2339

Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see difdisj 2335). Part of proof of Corollary 6K of [Enderton] p. 144.

Assertion

Ref	Expression
undif2	|- (A u. (B \ A)) = (A u. B)

Proof of Theorem undif2

Step	Hyp	Ref	Expression
1		uncom 2174	. 2 |- (A u. (B \ A)) = ((B \ A) u. A)
2		undif1 2338	. 2 |- ((B \ A) u. A) = (B u. A)
3		uncom 2174	. 2 |- (B u. A) = (A u. B)
4	1, 2, 3	3eqtr 1498	1 |- (A u. (B \ A)) = (A u. B)

Syntax hints:   = wceq 955   \ cdif 2042   u. cun 2043

This theorem is referenced by:  undif 2341  difex2 2874  oaabs 4249  undom 4431  unfi 4541  kmlem11 4762  infxpidmlem12 7542

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  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 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279

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