Theorem undifss 3303

Description: Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
undifss	|- ( A C_ B <-> ( A u. ( B \ A ) ) C_ B )

Proof of Theorem undifss

Step	Hyp	Ref	Expression
1		difss 3070	. . . 4 |- ( B \ A ) C_ B
2	1	jctr 298	. . 3 |- ( A C_ B -> ( A C_ B /\ ( B \ A ) C_ B ) )
3		unss 3117	. . 3 |- ( ( A C_ B /\ ( B \ A ) C_ B ) <-> ( A u. ( B \ A ) ) C_ B )
4	2, 3	sylib 127	. 2 |- ( A C_ B -> ( A u. ( B \ A ) ) C_ B )
5		ssun1 3106	. . 3 |- A C_ ( A u. ( B \ A ) )
6		sstr 2953	. . 3 |- ( ( A C_ ( A u. ( B \ A ) ) /\ ( A u. ( B \ A ) ) C_ B ) -> A C_ B )
7	5, 6	mpan 400	. 2 |- ( ( A u. ( B \ A ) ) C_ B -> A C_ B )
8	4, 7	impbii 117	1 |- ( A C_ B <-> ( A u. ( B \ A ) ) C_ B )

Syntax hints:   /\ wa 97   <-> wb 98   \ cdif 2914   u. cun 2915   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931

This theorem is referenced by:  difsnss  3510