Theorem undif2ss 3510

Description: Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
undif2ss	|- ( A u. ( B \ A ) ) C_ ( A u. B )

Proof of Theorem undif2ss

Step	Hyp	Ref	Expression
1		undif1ss 3509	. 2 |- ( ( B \ A ) u. A ) C_ ( B u. A )
2		uncom 3291	. 2 |- ( A u. ( B \ A ) ) = ( ( B \ A ) u. A )
3		uncom 3291	. 2 |- ( A u. B ) = ( B u. A )
4	1, 2, 3	3sstr4i 3208	1 |- ( A u. ( B \ A ) ) C_ ( A u. B )

Syntax hints:   \ cdif 3138   u. cun 3139   C_ wss 3141

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154

This theorem is referenced by: (None)