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Theorem tpid3 4701
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 4700 . 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 2108  Vcvv 3493  {ctp 4563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-un 3939  df-sn 4560  df-pr 4562  df-tp 4564
This theorem is referenced by:  wrdl3s3  14318  umgrwwlks2on  27728  ex-pss  28199  s3rn  30615  cyc3evpm  30785  sgnsf  30797  sgncl  31789  prodfzo03  31867  circlevma  31906  circlemethhgt  31907  hgt750lemg  31918  hgt750lemb  31920  hgt750lema  31921  hgt750leme  31922  tgoldbachgtde  31924  tgoldbachgt  31927  kur14lem7  32452  brtpid3  32946  rabren3dioph  39402  fourierdlem114  42495
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