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Theorem prid1g 3501
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 2056 . . 3 𝐴 = 𝐴
21orci 660 . 2 (𝐴 = 𝐴𝐴 = 𝐵)
3 elprg 3422 . 2 (𝐴𝑉 → (𝐴 ∈ {𝐴, 𝐵} ↔ (𝐴 = 𝐴𝐴 = 𝐵)))
42, 3mpbiri 161 1 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4  wo 639   = wceq 1259  wcel 1409  {cpr 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409
This theorem is referenced by:  prid2g  3502  prid1  3503  preqr1g  3564  opth1  4000  en2lp  4305  acexmidlemcase  5534  m1expcl2  9441
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