Theorem ssundifim 3608

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 939	. . . 4 |- ( ( x e. A -> ( x e. B \/ x e. C ) ) -> ( ( x e. A /\ -. x e. B ) -> x e. C ) )
2		elun 3370	. . . . 5 |- ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
3	2	imbi2i 226	. . . 4 |- ( ( x e. A -> x e. ( B u. C ) ) <-> ( x e. A -> ( x e. B \/ x e. C ) ) )
4		eldif 3229	. . . . 5 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
5	4	imbi1i 238	. . . 4 |- ( ( x e. ( A \ B ) -> x e. C ) <-> ( ( x e. A /\ -. x e. B ) -> x e. C ) )
6	1, 3, 5	3imtr4i 201	. . 3 |- ( ( x e. A -> x e. ( B u. C ) ) -> ( x e. ( A \ B ) -> x e. C ) )
7	6	alimi 1508	. 2 |- ( A. x ( x e. A -> x e. ( B u. C ) ) -> A. x ( x e. ( A \ B ) -> x e. C ) )
8		ssalel 3235	. 2 |- ( A C_ ( B u. C ) <-> A. x ( x e. A -> x e. ( B u. C ) ) )
9		ssalel 3235	. 2 |- ( ( A \ B ) C_ C <-> A. x ( x e. ( A \ B ) -> x e. C ) )
10	7, 8, 9	3imtr4i 201	1 |- ( A C_ ( B u. C ) -> ( A \ B ) C_ C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720   A.wal 1400   e. wcel 2209   \ cdif 3217   u. cun 3218   C_ wss 3220

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233

This theorem is referenced by:  hashfibclem  11260