Theorem ssdifss 2736

Description: Preservation of a subclass relationship by class difference.

Assertion

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

Proof of Theorem ssdifss

Step	Hyp	Ref	Expression
1		difss 2735	. 2 |- (A \ C) C_ A
2		sstr 2625	. 2 |- (((A \ C) C_ A /\ A C_ B) -> (A \ C) C_ B)
3	1, 2	mpan 759	1 |- (A C_ B -> (A \ C) C_ B)

Syntax hints:   -> wi 3   \ cdif 2590   C_ wss 2593

This theorem is referenced by:  unblem1 5633  xrsupss 7287  xrinfmss 7288  islp2 9023  subcld 10254  rcfpfillem6 14933  ufinffr 15578  fimax 15746  inficl 15757  lpss2 15842

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605

Copyright terms: Public domain