Theorem difss2 3309

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 3307	. 2 |- ( B \ C ) C_ B
3	1, 2	sstrdi 3213	1 |- ( A C_ ( B \ C ) -> A C_ B )

Syntax hints:   -> wi 4   \ cdif 3171   C_ wss 3174

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187

This theorem is referenced by:  difss2d  3310  ssdifsn  3772  sbthlem1  7085