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Theorem prssd 4573
Description: Deduction version of prssi 4572: 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 4572 . 2 ((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ⊆ 𝐶)
41, 2, 3syl2anc 579 1 (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2164  wss 3798  {cpr 4401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-un 3803  df-in 3805  df-ss 3812  df-sn 4400  df-pr 4402
This theorem is referenced by:  nehash2  13552  basprssdmsets  16295  dchrisum0re  25622  upgrex  26397  upgr1e  26418  uspgr1e  26548  eupth2lems  27611  pmtridf1o  30397  poimirlem9  33957  clsk1indlem4  39177  clsk1indlem1  39178  limsup10exlem  40793  meadjun  41464  line2  43314  line2y  43317
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