Theorem undifabs 3365

Description: Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)

Assertion

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

Proof of Theorem undifabs

Step	Hyp	Ref	Expression
1		ssid 3047	. . 3 |- A C_ A
2		difss 3129	. . 3 |- ( A \ B ) C_ A
3	1, 2	unssi 3178	. 2 |- ( A u. ( A \ B ) ) C_ A
4		ssun1 3166	. 2 |- A C_ ( A u. ( A \ B ) )
5	3, 4	eqssi 3044	1 |- ( A u. ( A \ B ) ) = A

Syntax hints:   = wceq 1290   \ cdif 2999   u. cun 3000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015

This theorem is referenced by: (None)