Theorem sscond 3218

Description: If A is contained in B, then ( C \ B ) is contained in ( C \ A ). Deduction form of sscon 3215. (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 3215	. 2 |- ( A C_ B -> ( C \ B ) C_ ( C \ A ) )
3	1, 2	syl 14	1 |- ( ph -> ( C \ B ) C_ ( C \ A ) )

Syntax hints:   -> wi 4   \ cdif 3073   C_ wss 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089

This theorem is referenced by:  ssdif2d  3220  setsresg  12036