Theorem tpid3 4741

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 4740	. 2 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
3	1, 2	ax-mp 5	1 ⊢ 𝐶 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∈ wcel 2149  Vcvv 3463  {ctp 4595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-sn 4592  df-pr 4594  df-tp 4596

This theorem is referenced by:  hash3tpb  14528  wrdl3s3  14995  sgncl  15130  usgrwwlks2on  30244  umgrwwlks2on  30245  ex-pss  30716  s3rnOLD  33203  cyc3evpm  33407  sgnsf  33419  prodfzo03  34931  circlevma  34970  circlemethhgt  34971  hgt750lemg  34982  hgt750lemb  34984  hgt750lema  34985  hgt750leme  34986  tgoldbachgtde  34988  tgoldbachgt  34991  kur14lem7  35599  brtpid3  36110  rabren3dioph  43427  oenord1ex  43927  fourierdlem114  46819  usgrexmpl1tri  48672  usgrexmpl2nb0  48678  usgrexmpl2nb1  48679  usgrexmpl2nb2  48680  usgrexmpl2nb3  48681  usgrexmpl2nb4  48682  usgrexmpl2nb5  48683  gpg3kgrtriex  48736