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Theorem ssdifss 4140
Description: Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
Assertion
Ref Expression
ssdifss (𝐴𝐵 → (𝐴𝐶) ⊆ 𝐵)

Proof of Theorem ssdifss
StepHypRef Expression
1 difss 4136 . 2 (𝐴𝐶) ⊆ 𝐴
2 sstr 3992 . 2 (((𝐴𝐶) ⊆ 𝐴𝐴𝐵) → (𝐴𝐶) ⊆ 𝐵)
31, 2mpan 690 1 (𝐴𝐵 → (𝐴𝐶) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  cdif 3948  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-ss 3968
This theorem is referenced by:  ssdifssd  4147  xrsupss  13351  xrinfmss  13352  rpnnen2lem12  16261  lpval  23147  lpdifsn  23151  islp2  23153  lpcls  23372  mblfinlem3  37666  mblfinlem4  37667  voliunnfl  37671  redvmptabs  42390  ssdifcl  43584  sssymdifcl  43585  supxrmnf2  45444  infxrpnf2  45474  fourierdlem102  46223  fourierdlem114  46235  lindslinindimp2lem4  48378  lindslinindsimp2lem5  48379  lindslinindsimp2  48380  lincresunit3  48398
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