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Theorem prid1g 3634
Description: An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
Assertion
Ref Expression
prid1g (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})

Proof of Theorem prid1g
StepHypRef Expression
1 eqid 2140 . . 3 𝐴 = 𝐴
21orci 721 . 2 (𝐴 = 𝐴𝐴 = 𝐵)
3 elprg 3551 . 2 (𝐴𝑉 → (𝐴 ∈ {𝐴, 𝐵} ↔ (𝐴 = 𝐴𝐴 = 𝐵)))
42, 3mpbiri 167 1 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4  wo 698   = wceq 1332  wcel 1481  {cpr 3532
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538
This theorem is referenced by:  prid2g  3635  prid1  3636  preqr1g  3700  opth1  4165  en2lp  4476  acexmidlemcase  5776  en2eqpr  6808  m1expcl2  10345  maxabslemval  11011  xrmaxiflemval  11050  xrmaxaddlem  11060  2strbasg  12097  2strbas1g  12100  coseq0negpitopi  12963
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