Theorem undif2ss 3544

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 3543	. 2 |- ( ( B \ A ) u. A ) C_ ( B u. A )
2		uncom 3325	. 2 |- ( A u. ( B \ A ) ) = ( ( B \ A ) u. A )
3		uncom 3325	. 2 |- ( A u. B ) = ( B u. A )
4	1, 2, 3	3sstr4i 3242	1 |- ( A u. ( B \ A ) ) C_ ( A u. B )

Syntax hints:   \ cdif 3171   u. cun 3172   C_ wss 3174

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 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187

This theorem is referenced by: (None)