MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssdifss Structured version   Visualization version   GIF version

Theorem ssdifss 3725
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 3721 . 2 (𝐴𝐶) ⊆ 𝐴
2 sstr 3596 . 2 (((𝐴𝐶) ⊆ 𝐴𝐴𝐵) → (𝐴𝐶) ⊆ 𝐵)
31, 2mpan 705 1 (𝐴𝐵 → (𝐴𝐶) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  cdif 3557  wss 3560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-dif 3563  df-in 3567  df-ss 3574
This theorem is referenced by:  ssdifssd  3732  xrsupss  12098  xrinfmss  12099  rpnnen2lem12  14898  lpval  20883  lpdifsn  20887  islp2  20889  lpcls  21108  mblfinlem3  33119  mblfinlem4  33120  voliunnfl  33124  ssdifcl  37396  sssymdifcl  37397  fourierdlem102  39762  fourierdlem114  39774  lindslinindimp2lem4  41568  lindslinindsimp2lem5  41569  lindslinindsimp2  41570  lincresunit3  41588
  Copyright terms: Public domain W3C validator