Theorem undif1ss 3512

Description: Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
undif1ss	|- ( ( A \ B ) u. B ) C_ ( A u. B )

Proof of Theorem undif1ss

Step	Hyp	Ref	Expression
1		difss 3276	. 2 |- ( A \ B ) C_ A
2		unss1 3319	. 2 |- ( ( A \ B ) C_ A -> ( ( A \ B ) u. B ) C_ ( A u. B ) )
3	1, 2	ax-mp 5	1 |- ( ( A \ B ) u. B ) C_ ( A u. B )

Syntax hints:   \ cdif 3141   u. cun 3142   C_ wss 3144

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157

This theorem is referenced by:  undif2ss  3513  pwundifss  4300