Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ssinss1d Structured version   Visualization version   GIF version

Theorem ssinss1d 38699
 Description: Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
ssinss1d.1 (𝜑𝐴𝐶)
Assertion
Ref Expression
ssinss1d (𝜑 → (𝐴𝐵) ⊆ 𝐶)

Proof of Theorem ssinss1d
StepHypRef Expression
1 ssinss1d.1 . 2 (𝜑𝐴𝐶)
2 ssinss1 3819 . 2 (𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)
31, 2syl 17 1 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∩ cin 3554   ⊆ wss 3555 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 3188  df-in 3562  df-ss 3569 This theorem is referenced by:  ssinss2d  38713  ovolsplit  39512  caragenuncllem  40033  carageniuncllem1  40042  ovnsplit  40169  vonvolmbllem  40181  vonvolmbl  40182
 Copyright terms: Public domain W3C validator