Theorem ssdifss 3257

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 3253	. 2 |- ( A \ C ) C_ A
2		sstr 3155	. 2 |- ( ( ( A \ C ) C_ A /\ A C_ B ) -> ( A \ C ) C_ B )
3	1, 2	mpan 422	1 |- ( A C_ B -> ( A \ C ) C_ B )

Syntax hints:   -> wi 4   \ 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:  ssdifssd  3265