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Theorem prprc2 4445
 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 4411 . 2 {𝐴, 𝐵} = {𝐵, 𝐴}
2 prprc1 4444 . 2 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴})
31, 2syl5eq 2806 1 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1632   ∈ wcel 2139  Vcvv 3340  {csn 4321  {cpr 4323 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-un 3720  df-nul 4059  df-sn 4322  df-pr 4324 This theorem is referenced by:  tpprceq3  4480  elpreqprlem  4546  prex  5058  indislem  21006  1to2vfriswmgr  27433  indispconn  31523  elsprel  42235
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