Theorem ssdifss 3337

Description: Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)

Assertion

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

Proof of Theorem ssdifss

Step	Hyp	Ref	Expression
1		difss 3333	. 2 |- ( A \ C ) C_ A
2		sstr 3235	. 2 |- ( ( ( A \ C ) C_ A /\ A C_ B ) -> ( A \ C ) C_ B )
3	1, 2	mpan 424	1 |- ( A C_ B -> ( A \ C ) C_ B )

Syntax hints:   -> wi 4   \ 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:  ssdifssd  3345