MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpid3 Structured version   Visualization version   GIF version

Theorem tpid3 4282
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 4280 . 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 1992  Vcvv 3191  {ctp 4157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-un 3565  df-sn 4154  df-pr 4156  df-tp 4158
This theorem is referenced by:  wrdl3s3  13634  umgrwwlks2on  26713  ex-pss  27133  sgnsf  29506  sgncl  30373  kur14lem7  30894  brtpid3  31305  rabren3dioph  36826  fourierdlem114  39712
  Copyright terms: Public domain W3C validator