Theorem ssundifim 3306

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 836	. . . 4 |- ( ( x e. A -> ( x e. B \/ x e. C ) ) -> ( ( x e. A /\ -. x e. B ) -> x e. C ) )
2		elun 3084	. . . . 5 |- ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
3	2	imbi2i 215	. . . 4 |- ( ( x e. A -> x e. ( B u. C ) ) <-> ( x e. A -> ( x e. B \/ x e. C ) ) )
4		eldif 2927	. . . . 5 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
5	4	imbi1i 227	. . . 4 |- ( ( x e. ( A \ B ) -> x e. C ) <-> ( ( x e. A /\ -. x e. B ) -> x e. C ) )
6	1, 3, 5	3imtr4i 190	. . 3 |- ( ( x e. A -> x e. ( B u. C ) ) -> ( x e. ( A \ B ) -> x e. C ) )
7	6	alimi 1344	. 2 |- ( A. x ( x e. A -> x e. ( B u. C ) ) -> A. x ( x e. ( A \ B ) -> x e. C ) )
8		dfss2 2934	. 2 |- ( A C_ ( B u. C ) <-> A. x ( x e. A -> x e. ( B u. C ) ) )
9		dfss2 2934	. 2 |- ( ( A \ B ) C_ C <-> A. x ( x e. ( A \ B ) -> x e. C ) )
10	7, 8, 9	3imtr4i 190	1 |- ( A C_ ( B u. C ) -> ( A \ B ) C_ C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   \/ wo 629   A.wal 1241   e. wcel 1393   \ 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: (None)