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Theorem prid2 3598
 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 3597 . 2 𝐵 ∈ {𝐵, 𝐴}
3 prcom 3567 . 2 {𝐵, 𝐴} = {𝐴, 𝐵}
42, 3eleqtri 2190 1 𝐵 ∈ {𝐴, 𝐵}
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1463  Vcvv 2658  {cpr 3496 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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502 This theorem is referenced by:  prel12  3666  opi2  4123  opeluu  4339  ontr2exmid  4408  onsucelsucexmid  4413  regexmidlemm  4415  ordtri2or2exmid  4454  dmrnssfld  4770  funopg  5125  acexmidlema  5731  acexmidlemcase  5735  acexmidlem2  5737  1lt2o  6305  2dom  6665  unfiexmid  6772  djuss  6921  exmidfodomrlemr  7022  exmidfodomrlemrALT  7023  exmidaclem  7028  cnelprrecn  7720  mnfxr  7786  sup3exmid  8672  m1expcl2  10255  bdop  12896  isomninnlem  13048
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