Theorem ssundif 3633

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		pm5.6 878	. . . 4 |- (((x e. A /\ -. x e. B) -> x e. C) <-> (x e. A -> (x e. B \/ x e. C)))
2		eldif 3221	. . . . 5 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
3	2	imbi1i 315	. . . 4 |- ((x e. (A \ B) -> x e. C) <-> ((x e. A /\ -. x e. B) -> x e. C))
4		elun 3220	. . . . 5 |- (x e. (B u. C) <-> (x e. B \/ x e. C))
5	4	imbi2i 303	. . . 4 |- ((x e. A -> x e. (B u. C)) <-> (x e. A -> (x e. B \/ x e. C)))
6	1, 3, 5	3bitr4ri 269	. . 3 |- ((x e. A -> x e. (B u. C)) <-> (x e. (A \ B) -> x e. C))
7	6	albii 1566	. 2 |- (A.x(x e. A -> x e. (B u. C)) <-> A.x(x e. (A \ B) -> x e. C))
8		dfss2 3262	. 2 |- (A C_ (B u. C) <-> A.x(x e. A -> x e. (B u. C)))
9		dfss2 3262	. 2 |- ((A \ B) C_ C <-> A.x(x e. (A \ B) -> x e. C))
10	7, 8, 9	3bitr4i 268	1 |- (A C_ (B u. C) <-> (A \ B) C_ C)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358  A.wal 1540   e. wcel 1710   \ cdif 3206   u. cun 3207   C_ wss 3257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259

This theorem is referenced by:  difcom  3634  uneqdifeq  3638  ssunsn2  3865  pwadjoin  4119