Theorem ssdifss 3294

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 3290	. 2 |- ( A \ C ) C_ A
2		sstr 3192	. 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 3154   C_ wss 3157

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170

This theorem is referenced by:  ssdifssd  3302