Theorem undifabs 3541

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 3217	. . 3 |- A C_ A
2		difss 3303	. . 3 |- ( A \ B ) C_ A
3	1, 2	unssi 3352	. 2 |- ( A u. ( A \ B ) ) C_ A
4		ssun1 3340	. 2 |- A C_ ( A u. ( A \ B ) )
5	3, 4	eqssi 3213	1 |- ( A u. ( A \ B ) ) = A

Syntax hints:   = wceq 1373   \ cdif 3167   u. cun 3168

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183

This theorem is referenced by:  exmid1stab  4260