Theorem undifabs 3536

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 3212	. . 3 |- A C_ A
2		difss 3298	. . 3 |- ( A \ B ) C_ A
3	1, 2	unssi 3347	. 2 |- ( A u. ( A \ B ) ) C_ A
4		ssun1 3335	. 2 |- A C_ ( A u. ( A \ B ) )
5	3, 4	eqssi 3208	1 |- ( A u. ( A \ B ) ) = A

Syntax hints:   = wceq 1372   \ cdif 3162   u. cun 3163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178

This theorem is referenced by:  exmid1stab  4251