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Theorem prssd 4821
Description: Deduction version of prssi 4820: 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 4820 . 2 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ⊆ 𝐶)
41, 2, 3syl2anc 583 1 (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  wss 3944  {cpr 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3471  df-un 3949  df-in 3951  df-ss 3961  df-sn 4625  df-pr 4627
This theorem is referenced by:  fpr2g  7217  f1prex  7287  fveqf1o  7306  fr3nr  7768  en2eqpr  10024  en2eleq  10025  r0weon  10029  wuncval2  10764  nehash2  14461  1idssfct  16644  basprssdmsets  17186  mrcun  17595  joinval2  18366  meetval2  18380  0idnsgd  19119  pmtrprfv  19401  pmtrprfv3  19402  symggen  19418  pmtr3ncomlem1  19421  psgnunilem1  19441  lspprcl  20855  lsptpcl  20856  lspprss  20869  lspprid1  20874  lsppratlem2  21029  lsppratlem3  21030  lsppratlem4  21031  drngnidl  21131  drnglpir  21215  mdetralt  22503  topgele  22825  pptbas  22904  isconn2  23311  xpsdsval  24280  itgioo  25738  wilthlem2  26994  perfectlem2  27156  upgrex  28898  upgr1e  28919  uspgr1e  29050  eupth2lems  30041  s2f1  32662  pmtrcnel  32806  pmtrcnel2  32807  pmtridf1o  32809  cycpm2tr  32834  cyc3co2  32855  cyc3evpm  32865  cyc3genpmlem  32866  cyc3conja  32872  linds2eq  33090  poimirlem9  37091  clsk1indlem4  43446  clsk1indlem1  43447  mnuprssd  43678  mnuprdlem4  43684  limsup10exlem  45132  meadjun  45822  line2  47797  line2y  47800  lubprlem  47953  joindm3  47960  meetdm3  47962  toplatjoin  47985  toplatmeet  47986
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