Theorem undifss 3474

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 3233	. . . 4 |- ( B \ A ) C_ B
2	1	jctr 313	. . 3 |- ( A C_ B -> ( A C_ B /\ ( B \ A ) C_ B ) )
3		unss 3281	. . 3 |- ( ( A C_ B /\ ( B \ A ) C_ B ) <-> ( A u. ( B \ A ) ) C_ B )
4	2, 3	sylib 121	. 2 |- ( A C_ B -> ( A u. ( B \ A ) ) C_ B )
5		ssun1 3270	. . 3 |- A C_ ( A u. ( B \ A ) )
6		sstr 3136	. . 3 |- ( ( A C_ ( A u. ( B \ A ) ) /\ ( A u. ( B \ A ) ) C_ B ) -> A C_ B )
7	5, 6	mpan 421	. 2 |- ( ( A u. ( B \ A ) ) C_ B -> A C_ B )
8	4, 7	impbii 125	1 |- ( A C_ B <-> ( A u. ( B \ A ) ) C_ B )

Syntax hints:   /\ wa 103   <-> wb 104   \ cdif 3099   u. cun 3100   C_ wss 3102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115

This theorem is referenced by:  difsnss  3702  exmidundif  4167  exmidundifim  4168  undifdcss  6867