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Theorem prid2 3600
Description: An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
prid2.1 𝐵 ∈ V
Assertion
Ref Expression
prid2 𝐵 ∈ {𝐴, 𝐵}

Proof of Theorem prid2
StepHypRef Expression
1 prid2.1 . . 3 𝐵 ∈ V
21prid1 3599 . 2 𝐵 ∈ {𝐵, 𝐴}
3 prcom 3569 . 2 {𝐵, 𝐴} = {𝐴, 𝐵}
42, 3eleqtri 2192 1 𝐵 ∈ {𝐴, 𝐵}
Colors of variables: wff set class
Syntax hints:  wcel 1465  Vcvv 2660  {cpr 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504
This theorem is referenced by:  prel12  3668  opi2  4125  opeluu  4341  ontr2exmid  4410  onsucelsucexmid  4415  regexmidlemm  4417  ordtri2or2exmid  4456  dmrnssfld  4772  funopg  5127  acexmidlema  5733  acexmidlemcase  5737  acexmidlem2  5739  1lt2o  6307  2dom  6667  unfiexmid  6774  djuss  6923  exmidonfinlem  7017  exmidfodomrlemr  7026  exmidfodomrlemrALT  7027  exmidaclem  7032  cnelprrecn  7724  mnfxr  7790  sup3exmid  8683  m1expcl2  10283  bdop  13000  isomninnlem  13152
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