Theorem undifabs 3571

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 3247	. . 3 |- A C_ A
2		difss 3333	. . 3 |- ( A \ B ) C_ A
3	1, 2	unssi 3382	. 2 |- ( A u. ( A \ B ) ) C_ A
4		ssun1 3370	. 2 |- A C_ ( A u. ( A \ B ) )
5	3, 4	eqssi 3243	1 |- ( A u. ( A \ B ) ) = A

Syntax hints:   = wceq 1397   \ cdif 3197   u. cun 3198

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213

This theorem is referenced by:  exmid1stab  4298