Theorem difss2 3739

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 22	. 2 |- ( A C_ ( B \ C ) -> A C_ ( B \ C ) )
2		difss 3737	. 2 |- ( B \ C ) C_ B
3	1, 2	syl6ss 3615	1 |- ( A C_ ( B \ C ) -> A C_ B )

Syntax hints:   -> wi 4   \ cdif 3571   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588

This theorem is referenced by:  difss2d  3740  sbthlem1  8070  bcthlem2  23122  ismblfin  33450