Theorem tpid3 4727

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 4726	. 2 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
3	1, 2	ax-mp 5	1 ⊢ 𝐶 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∈ wcel 2109  Vcvv 3438  {ctp 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3440  df-un 3910  df-sn 4580  df-pr 4582  df-tp 4584

This theorem is referenced by:  hash3tpb  14420  wrdl3s3  14887  umgrwwlks2on  29920  ex-pss  30390  sgncl  32789  s3rnOLD  32900  cyc3evpm  33105  sgnsf  33117  prodfzo03  34573  circlevma  34612  circlemethhgt  34613  hgt750lemg  34624  hgt750lemb  34626  hgt750lema  34627  hgt750leme  34628  tgoldbachgtde  34630  tgoldbachgt  34633  kur14lem7  35187  brtpid3  35698  rabren3dioph  42791  oenord1ex  43291  fourierdlem114  46205  usgrexmpl1tri  48013  usgrexmpl2nb0  48019  usgrexmpl2nb1  48020  usgrexmpl2nb2  48021  usgrexmpl2nb3  48022  usgrexmpl2nb4  48023  usgrexmpl2nb5  48024  gpg3kgrtriex  48077