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Theorem tpid1 4246
 Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypothesis
Ref Expression
tpid1.1 𝐴 ∈ V
Assertion
Ref Expression
tpid1 𝐴 ∈ {𝐴, 𝐵, 𝐶}

Proof of Theorem tpid1
StepHypRef Expression
1 eqid 2610 . . 3 𝐴 = 𝐴
213mix1i 1226 . 2 (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶)
3 tpid1.1 . . 3 𝐴 ∈ V
43eltp 4177 . 2 (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶))
52, 4mpbir 220 1 𝐴 ∈ {𝐴, 𝐵, 𝐶}
 Colors of variables: wff setvar class Syntax hints:   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {ctp 4129 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128  df-tp 4130 This theorem is referenced by:  tpnz  4256  wrdl3s3  13553  2pthlem2  26126  usgra2adedgwlkonALT  26144  sgnsf  29060  sgncl  29927  kur14lem7  30448  kur14lem9  30450  brtpid1  30857  rabren3dioph  36397  fourierdlem102  39101  fourierdlem114  39113  etransclem48  39175  umgrwwlks2on  41161
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