Theorem ssundif 3910

Description: A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.)

Assertion

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

Proof of Theorem ssundif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm5.6 910	. . . 4 |- ( ( ( x e. A /\ -. x e. B ) -> x e. C ) <-> ( x e. A -> ( x e. B \/ x e. C ) ) )
2		eldif 3486	. . . . 5 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
3	2	imbi1i 325	. . . 4 |- ( ( x e. ( A \ B ) -> x e. C ) <-> ( ( x e. A /\ -. x e. B ) -> x e. C ) )
4		elun 3645	. . . . 5 |- ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
5	4	imbi2i 312	. . . 4 |- ( ( x e. A -> x e. ( B u. C ) ) <-> ( x e. A -> ( x e. B \/ x e. C ) ) )
6	1, 3, 5	3bitr4ri 278	. . 3 |- ( ( x e. A -> x e. ( B u. C ) ) <-> ( x e. ( A \ B ) -> x e. C ) )
7	6	albii 1620	. 2 |- ( A. x ( x e. A -> x e. ( B u. C ) ) <-> A. x ( x e. ( A \ B ) -> x e. C ) )
8		dfss2 3493	. 2 |- ( A C_ ( B u. C ) <-> A. x ( x e. A -> x e. ( B u. C ) ) )
9		dfss2 3493	. 2 |- ( ( A \ B ) C_ C <-> A. x ( x e. ( A \ B ) -> x e. C ) )
10	7, 8, 9	3bitr4i 277	1 |- ( A C_ ( B u. C ) <-> ( A \ B ) C_ C )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   A.wal 1377   e. wcel 1767   \ cdif 3473   u. cun 3474   C_ wss 3476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490

This theorem is referenced by:  difcom  3911  uneqdifeq  3915  ssunsn2  4186  elpwun  6597  soex  6727  ressuppssdif  6921  frfi  7765  cantnfp1lem3  8099  cantnfp1lem3OLD  8125  dfacfin7  8779  zornn0g  8885  fpwwe2lem13  9020  hashbclem  12467  incexclem  13611  ramub1lem1  14403  lpcls  19659  cmpcld  19696  alexsubALTlem3  20312  restmetu  20853  uniiccdif  21750  abelthlem2  22589  abelthlem3  22590