Theorem ssdif 3282

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 3161	. . . 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 3150	. . 3 |- ( x e. ( A \ C ) <-> ( x e. A /\ -. x e. C ) )
4		eldif 3150	. . 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 3173	1 |- ( A C_ B -> ( A \ C ) C_ ( B \ C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2158   \ cdif 3138   C_ wss 3141

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-in 3147  df-ss 3154

This theorem is referenced by:  ssdifd  3283  phpm  6879  difinfinf  7114