Theorem undifabs 3491

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 3167	. . 3 |- A C_ A
2		difss 3253	. . 3 |- ( A \ B ) C_ A
3	1, 2	unssi 3302	. 2 |- ( A u. ( A \ B ) ) C_ A
4		ssun1 3290	. 2 |- A C_ ( A u. ( A \ B ) )
5	3, 4	eqssi 3163	1 |- ( A u. ( A \ B ) ) = A

Syntax hints:   = wceq 1348   \ cdif 3118   u. cun 3119

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134

This theorem is referenced by:  exmid1stab  14033