Theorem ssdif 3211

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 3091	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 334	. . 3 |- ( A C_ B -> ( ( x e. A /\ -. x e. C ) -> ( x e. B /\ -. x e. C ) ) )
3		eldif 3080	. . 3 |- ( x e. ( A \ C ) <-> ( x e. A /\ -. x e. C ) )
4		eldif 3080	. . 3 |- ( x e. ( B \ C ) <-> ( x e. B /\ -. x e. C ) )
5	2, 3, 4	3imtr4g 204	. 2 |- ( A C_ B -> ( x e. ( A \ C ) -> x e. ( B \ C ) ) )
6	5	ssrdv 3103	1 |- ( A C_ B -> ( A \ C ) C_ ( B \ C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 1480   \ cdif 3068   C_ wss 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084

This theorem is referenced by:  ssdifd  3212  phpm  6759  difinfinf  6986