Theorem undifabs 4045

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		undif3 3888	. 2 |- ( A u. ( A \ B ) ) = ( ( A u. A ) \ ( B \ A ) )
2		unidm 3756	. . 3 |- ( A u. A ) = A
3	2	difeq1i 3724	. 2 |- ( ( A u. A ) \ ( B \ A ) ) = ( A \ ( B \ A ) )
4		difdif 3736	. 2 |- ( A \ ( B \ A ) ) = A
5	1, 3, 4	3eqtri 2648	1 |- ( A u. ( A \ B ) ) = A

Syntax hints:   = wceq 1483   \ cdif 3571   u. cun 3572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579

This theorem is referenced by:  dfif5  4102  indifundif  29356