Theorem undifabs 3444

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 3122	. . 3 |- A C_ A
2		difss 3207	. . 3 |- ( A \ B ) C_ A
3	1, 2	unssi 3256	. 2 |- ( A u. ( A \ B ) ) C_ A
4		ssun1 3244	. 2 |- A C_ ( A u. ( A \ B ) )
5	3, 4	eqssi 3118	1 |- ( A u. ( A \ B ) ) = A

Syntax hints:   = wceq 1332   \ cdif 3073   u. cun 3074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089

This theorem is referenced by:  exmid1stab  13368