Theorem undif1ss 3525

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 3289	. 2 |- ( A \ B ) C_ A
2		unss1 3332	. 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 3154   u. cun 3155   C_ wss 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170

This theorem is referenced by:  undif2ss  3526  pwundifss  4320