Theorem undif2ss 3377

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 3376	. 2 |- ( ( B \ A ) u. A ) C_ ( B u. A )
2		uncom 3159	. 2 |- ( A u. ( B \ A ) ) = ( ( B \ A ) u. A )
3		uncom 3159	. 2 |- ( A u. B ) = ( B u. A )
4	1, 2, 3	3sstr4i 3080	1 |- ( A u. ( B \ A ) ) C_ ( A u. B )

Syntax hints:   \ cdif 3010   u. cun 3011   C_ wss 3013

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026

This theorem is referenced by: (None)