Theorem sscond 3747

Description: If A is contained in B, then ( C \ B ) is contained in ( C \ A ). Deduction form of sscon 3744. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
ssdifd.1	|- ( ph -> A C_ B )

Assertion

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

Proof of Theorem sscond

Step	Hyp	Ref	Expression
1		ssdifd.1	. 2 |- ( ph -> A C_ B )
2		sscon 3744	. 2 |- ( A C_ B -> ( C \ B ) C_ ( C \ A ) )
3	1, 2	syl 17	1 |- ( ph -> ( C \ B ) C_ ( C \ A ) )

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:  ssdif2d  3749  eldifeldifsn  4342  fin23lem26  9147  isercoll2  14399  fctop  20808  ntrss  20859  iunconnlem  21230  clsconn  21233  regr1lem  21542  blcld  22310  rrxdstprj1  23192  voliunlem1  23318  carsgclctunlem2  30381  salexct  40552  meaiininclem  40700  carageniuncllem2  40736