Theorem undif2ss 3536

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 3535	. 2 |- ( ( B \ A ) u. A ) C_ ( B u. A )
2		uncom 3317	. 2 |- ( A u. ( B \ A ) ) = ( ( B \ A ) u. A )
3		uncom 3317	. 2 |- ( A u. B ) = ( B u. A )
4	1, 2, 3	3sstr4i 3234	1 |- ( A u. ( B \ A ) ) C_ ( A u. B )

Syntax hints:   \ cdif 3163   u. cun 3164   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179

This theorem is referenced by: (None)