Theorem undifss 3549

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 3307	. . . 4 |- ( B \ A ) C_ B
2	1	jctr 315	. . 3 |- ( A C_ B -> ( A C_ B /\ ( B \ A ) C_ B ) )
3		unss 3355	. . 3 |- ( ( A C_ B /\ ( B \ A ) C_ B ) <-> ( A u. ( B \ A ) ) C_ B )
4	2, 3	sylib 122	. 2 |- ( A C_ B -> ( A u. ( B \ A ) ) C_ B )
5		ssun1 3344	. . 3 |- A C_ ( A u. ( B \ A ) )
6		sstr 3209	. . 3 |- ( ( A C_ ( A u. ( B \ A ) ) /\ ( A u. ( B \ A ) ) C_ B ) -> A C_ B )
7	5, 6	mpan 424	. 2 |- ( ( A u. ( B \ A ) ) C_ B -> A C_ B )
8	4, 7	impbii 126	1 |- ( A C_ B <-> ( A u. ( B \ A ) ) C_ B )

Syntax hints:   /\ wa 104   <-> wb 105   \ cdif 3171   u. cun 3172   C_ wss 3174

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187

This theorem is referenced by:  difsnss  3790  exmidundif  4266  exmidundifim  4267  undifdcss  7046