Theorem tpid3 4729

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 4728	. 2 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
3	1, 2	ax-mp 5	1 ⊢ 𝐶 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∈ wcel 2141  Vcvv 3453  {ctp 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3907  df-sn 4580  df-pr 4582  df-tp 4584

This theorem is referenced by:  hash3tpb  14502  wrdl3s3  14969  sgncl  15103  usgrwwlks2on  30115  umgrwwlks2on  30116  ex-pss  30587  s3rnOLD  33085  cyc3evpm  33291  sgnsf  33303  prodfzo03  34858  circlevma  34897  circlemethhgt  34898  hgt750lemg  34909  hgt750lemb  34911  hgt750lema  34912  hgt750leme  34913  tgoldbachgtde  34915  tgoldbachgt  34918  kur14lem7  35523  brtpid3  36034  rabren3dioph  43353  oenord1ex  43853  fourierdlem114  46755  usgrexmpl1tri  48608  usgrexmpl2nb0  48614  usgrexmpl2nb1  48615  usgrexmpl2nb2  48616  usgrexmpl2nb3  48617  usgrexmpl2nb4  48618  usgrexmpl2nb5  48619  gpg3kgrtriex  48672