Theorem ssdif 3262

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 3141	. . . 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 3130	. . 3 |- ( x e. ( A \ C ) <-> ( x e. A /\ -. x e. C ) )
4		eldif 3130	. . 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 3153	1 |- ( A C_ B -> ( A \ C ) C_ ( B \ C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 2141   \ cdif 3118   C_ wss 3121

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134

This theorem is referenced by:  ssdifd  3263  phpm  6843  difinfinf  7078