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Theorem prprc2 4532
Description: A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
prprc2 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})

Proof of Theorem prprc2
StepHypRef Expression
1 prcom 4498 . 2 {𝐴, 𝐵} = {𝐵, 𝐴}
2 prprc1 4531 . 2 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴})
31, 2syl5eq 2825 1 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1601  wcel 2106  Vcvv 3397  {csn 4397  {cpr 4399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-v 3399  df-dif 3794  df-un 3796  df-nul 4141  df-sn 4398  df-pr 4400
This theorem is referenced by:  tpprceq3  4566  elpreqprlem  4629  prex  5141  indislem  21212  1to2vfriswmgr  27701  indispconn  31829  elsprel  42406
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