Theorem ssdif 2740

Description: Difference law for subsets.

Assertion

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

Proof of Theorem ssdif

Step	Hyp	Ref	Expression
1		ssel 2615	. . . 4 |- (A C_ B -> (x e. A -> x e. B))
2	1	anim1d 619	. . 3 |- (A C_ B -> ((x e. A /\ -. x e. C) -> (x e. B /\ -. x e. C)))
3		eldif 2609	. . 3 |- (x e. (A \ C) <-> (x e. A /\ -. x e. C))
4		eldif 2609	. . 3 |- (x e. (B \ C) <-> (x e. B /\ -. x e. C))
5	2, 3, 4	3imtr4g 612	. 2 |- (A C_ B -> (x e. (A \ C) -> x e. (B \ C)))
6	5	ssrdv 2622	1 |- (A C_ B -> (A \ C) C_ (B \ C))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   e. wcel 1300   \ cdif 2590   C_ wss 2593

This theorem is referenced by:  sspr 3144  php 5607  pssnn 5628  frfi 15771  lpss2 15842  divrngidl 16176

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605

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