Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  prprc2 Structured version   Visualization version   GIF version

Theorem prprc2 4244
 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 4211 . 2 {𝐴, 𝐵} = {𝐵, 𝐴}
2 prprc1 4243 . 2 𝐵 ∈ V → {𝐵, 𝐴} = {𝐴})
31, 2syl5eq 2656 1 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  {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-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-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128 This theorem is referenced by:  tpprceq3  4276  elpreqprlem  4333  prex  4836  indislem  20614  usgraedgprv  25905  1to2vfriswmgra  26533  indispcon  30470  1to2vfriswmgr  41449
 Copyright terms: Public domain W3C validator