Theorem tpid3 4777

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

Step	Hyp	Ref	Expression
1		tpid3.1	. 2 ⊢ 𝐶 ∈ V
2		tpid3g 4776	. 2 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
3	1, 2	ax-mp 5	1 ⊢ 𝐶 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∈ wcel 2107  Vcvv 3475  {ctp 4632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3953  df-sn 4629  df-pr 4631  df-tp 4633

This theorem is referenced by:  wrdl3s3  14910  umgrwwlks2on  29201  ex-pss  29671  s3rn  32100  cyc3evpm  32297  sgnsf  32309  sgncl  33526  prodfzo03  33604  circlevma  33643  circlemethhgt  33644  hgt750lemg  33655  hgt750lemb  33657  hgt750lema  33658  hgt750leme  33659  tgoldbachgtde  33661  tgoldbachgt  33664  kur14lem7  34192  brtpid3  34681  rabren3dioph  41539  oenord1ex  42051  fourierdlem114  44923