Theorem tpid3 4778

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 4777	. 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 4633

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 3954  df-sn 4630  df-pr 4632  df-tp 4634

This theorem is referenced by:  wrdl3s3  14913  umgrwwlks2on  29211  ex-pss  29681  s3rn  32112  cyc3evpm  32309  sgnsf  32321  sgncl  33537  prodfzo03  33615  circlevma  33654  circlemethhgt  33655  hgt750lemg  33666  hgt750lemb  33668  hgt750lema  33669  hgt750leme  33670  tgoldbachgtde  33672  tgoldbachgt  33675  kur14lem7  34203  brtpid3  34692  rabren3dioph  41553  oenord1ex  42065  fourierdlem114  44936