Theorem undifss 3541

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 3299	. . . 4 |- ( B \ A ) C_ B
2	1	jctr 315	. . 3 |- ( A C_ B -> ( A C_ B /\ ( B \ A ) C_ B ) )
3		unss 3347	. . 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 3336	. . 3 |- A C_ ( A u. ( B \ A ) )
6		sstr 3201	. . 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 3163   u. cun 3164   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179

This theorem is referenced by:  difsnss  3779  exmidundif  4250  exmidundifim  4251  undifdcss  7020