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Theorem prid2 3544
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 3543 . 2 𝐵 ∈ {𝐵, 𝐴}
3 prcom 3513 . 2 {𝐵, 𝐴} = {𝐴, 𝐵}
42, 3eleqtri 2162 1 𝐵 ∈ {𝐴, 𝐵}
Colors of variables: wff set class
Syntax hints:  wcel 1438  Vcvv 2619  {cpr 3442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448
This theorem is referenced by:  prel12  3610  opi2  4051  opeluu  4263  ontr2exmid  4331  onsucelsucexmid  4336  regexmidlemm  4338  ordtri2or2exmid  4377  dmrnssfld  4684  funopg  5034  acexmidlema  5625  acexmidlemcase  5629  acexmidlem2  5631  2dom  6502  unfiexmid  6608  djuss  6740  fodjuomnilemf  6779  exmidfodomrlemr  6807  exmidfodomrlemrALT  6808  cnelprrecn  7457  mnfxr  7523  m1expcl2  9942  bdop  11412  1lt2o  11532
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