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

Theorem prssd 4294
Description: Deduction version of prssi 4293: A pair of elements of a class is a subset of the class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
prssd.1 (𝜑𝐴𝐶)
prssd.2 (𝜑𝐵𝐶)
Assertion
Ref Expression
prssd (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)

Proof of Theorem prssd
StepHypRef Expression
1 prssd.1 . 2 (𝜑𝐴𝐶)
2 prssd.2 . 2 (𝜑𝐵𝐶)
3 prssi 4293 . 2 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ⊆ 𝐶)
41, 2, 3syl2anc 691 1 (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977  wss 3540  {cpr 4127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-pr 4128
This theorem is referenced by:  nehash2  13113  dchrisum0re  25002  uhgrstrrepelem  25744  upgrex  25759  upgr1e  25779  poimirlem9  32588  clsk1indlem4  37362  clsk1indlem1  37363  meadjun  39355  uspgr1e  40470  eupth2lems  41406
  Copyright terms: Public domain W3C validator