Theorem undifabs 3523

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 3199	. . 3 |- A C_ A
2		difss 3285	. . 3 |- ( A \ B ) C_ A
3	1, 2	unssi 3334	. 2 |- ( A u. ( A \ B ) ) C_ A
4		ssun1 3322	. 2 |- A C_ ( A u. ( A \ B ) )
5	3, 4	eqssi 3195	1 |- ( A u. ( A \ B ) ) = A

Syntax hints:   = wceq 1364   \ cdif 3150   u. cun 3151

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166

This theorem is referenced by:  exmid1stab  4237