MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difss2 Structured version   Visualization version   GIF version

Theorem difss2 4126
Description: If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
difss2 (𝐴 ⊆ (𝐵𝐶) → 𝐴𝐵)

Proof of Theorem difss2
StepHypRef Expression
1 id 22 . 2 (𝐴 ⊆ (𝐵𝐶) → 𝐴 ⊆ (𝐵𝐶))
2 difss 4124 . 2 (𝐵𝐶) ⊆ 𝐵
31, 2sstrdi 3987 1 (𝐴 ⊆ (𝐵𝐶) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  cdif 3938  wss 3941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3472  df-dif 3944  df-in 3948  df-ss 3958
This theorem is referenced by:  difss2d  4127  ssdifsn  4781  sbthlem1  9063  bcthlem2  24766  ismblfin  36317
  Copyright terms: Public domain W3C validator