Theorem difss2 3264

Description: If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)

Assertion

Ref	Expression
difss2	|- ( A C_ ( B \ C ) -> A C_ B )

Proof of Theorem difss2

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( A C_ ( B \ C ) -> A C_ ( B \ C ) )
2		difss 3262	. 2 |- ( B \ C ) C_ B
3	1, 2	sstrdi 3168	1 |- ( A C_ ( B \ C ) -> A C_ B )

Syntax hints:   -> wi 4   \ cdif 3127   C_ wss 3130

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-dif 3132  df-in 3136  df-ss 3143

This theorem is referenced by:  difss2d  3265  ssdifsn  3721  sbthlem1  6956