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Theorem prid1g 3593
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 2115 . . 3 𝐴 = 𝐴
21orci 703 . 2 (𝐴 = 𝐴𝐴 = 𝐵)
3 elprg 3513 . 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 680   = wceq 1314  wcel 1463  {cpr 3494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 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 2244  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500
This theorem is referenced by:  prid2g  3594  prid1  3595  preqr1g  3659  opth1  4118  en2lp  4429  acexmidlemcase  5723  en2eqpr  6754  m1expcl2  10205  maxabslemval  10869  xrmaxiflemval  10908  xrmaxaddlem  10918  2strbasg  11900  2strbas1g  11903
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