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Theorem tpid3 4250
 Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by JJ, 30-Apr-2021.)
Hypothesis
Ref Expression
tpid3.1 𝐶 ∈ V
Assertion
Ref Expression
tpid3 𝐶 ∈ {𝐴, 𝐵, 𝐶}

Proof of Theorem tpid3
StepHypRef Expression
1 tpid3.1 . 2 𝐶 ∈ V
2 tpid3g 4248 . 2 (𝐶 ∈ V → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
31, 2ax-mp 5 1 𝐶 ∈ {𝐴, 𝐵, 𝐶}
 Colors of variables: wff setvar class Syntax hints:   ∈ 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:  wrdl3s3  13553  2pthlem2  26126  usgra2adedgwlkonALT  26144  ex-pss  26677  sgnsf  29060  sgncl  29927  kur14lem7  30448  brtpid3  30859  rabren3dioph  36397  fourierdlem114  39113  umgrwwlks2on  41161
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