Theorem ssdif 3342

Description: Difference law for subsets. (Contributed by NM, 28-May-1998.)

Assertion

Ref	Expression
ssdif	|- ( A C_ B -> ( A \ C ) C_ ( B \ C ) )

Proof of Theorem ssdif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3221	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 336	. . 3 |- ( A C_ B -> ( ( x e. A /\ -. x e. C ) -> ( x e. B /\ -. x e. C ) ) )
3		eldif 3209	. . 3 |- ( x e. ( A \ C ) <-> ( x e. A /\ -. x e. C ) )
4		eldif 3209	. . 3 |- ( x e. ( B \ C ) <-> ( x e. B /\ -. x e. C ) )
5	2, 3, 4	3imtr4g 205	. 2 |- ( A C_ B -> ( x e. ( A \ C ) -> x e. ( B \ C ) ) )
6	5	ssrdv 3233	1 |- ( A C_ B -> ( A \ C ) C_ ( B \ C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2202   \ cdif 3197   C_ wss 3200

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213

This theorem is referenced by:  ssdifd  3343  phpm  7052  difinfinf  7300