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

Theorem difss2 3723
 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 3721 . 2 (𝐵𝐶) ⊆ 𝐵
31, 2syl6ss 3600 1 (𝐴 ⊆ (𝐵𝐶) → 𝐴𝐵)
 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:  difss2d  3724  sbthlem1  8030  bcthlem2  23062  ismblfin  33121
 Copyright terms: Public domain W3C validator