Theorem ssundifim 3477

Description: A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
ssundifim	|- ( A C_ ( B u. C ) -> ( A \ B ) C_ C )

Proof of Theorem ssundifim

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm5.6r 913	. . . 4 |- ( ( x e. A -> ( x e. B \/ x e. C ) ) -> ( ( x e. A /\ -. x e. B ) -> x e. C ) )
2		elun 3248	. . . . 5 |- ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
3	2	imbi2i 225	. . . 4 |- ( ( x e. A -> x e. ( B u. C ) ) <-> ( x e. A -> ( x e. B \/ x e. C ) ) )
4		eldif 3111	. . . . 5 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
5	4	imbi1i 237	. . . 4 |- ( ( x e. ( A \ B ) -> x e. C ) <-> ( ( x e. A /\ -. x e. B ) -> x e. C ) )
6	1, 3, 5	3imtr4i 200	. . 3 |- ( ( x e. A -> x e. ( B u. C ) ) -> ( x e. ( A \ B ) -> x e. C ) )
7	6	alimi 1435	. 2 |- ( A. x ( x e. A -> x e. ( B u. C ) ) -> A. x ( x e. ( A \ B ) -> x e. C ) )
8		dfss2 3117	. 2 |- ( A C_ ( B u. C ) <-> A. x ( x e. A -> x e. ( B u. C ) ) )
9		dfss2 3117	. 2 |- ( ( A \ B ) C_ C <-> A. x ( x e. ( A \ B ) -> x e. C ) )
10	7, 8, 9	3imtr4i 200	1 |- ( A C_ ( B u. C ) -> ( A \ B ) C_ C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698   A.wal 1333   e. wcel 2128   \ 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: (None)