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

Theorem ssinss1d 39735
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 3990 . 2 (𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)
31, 2syl 17 1 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3722  wss 3723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-ss 3737
This theorem is referenced by:  ssinss2d  39749  ovolsplit  40722  caragenuncllem  41246  carageniuncllem1  41255  ovnsplit  41382  vonvolmbllem  41394  vonvolmbl  41395
  Copyright terms: Public domain W3C validator