Theorem tpid3 4730

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 4729	. 2 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
3	1, 2	ax-mp 5	1 ⊢ 𝐶 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∈ wcel 2113  Vcvv 3440  {ctp 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-un 3906  df-sn 4581  df-pr 4583  df-tp 4585

This theorem is referenced by:  hash3tpb  14418  wrdl3s3  14885  usgrwwlks2on  30031  umgrwwlks2on  30032  ex-pss  30503  sgncl  32912  s3rnOLD  33028  cyc3evpm  33232  sgnsf  33244  prodfzo03  34760  circlevma  34799  circlemethhgt  34800  hgt750lemg  34811  hgt750lemb  34813  hgt750lema  34814  hgt750leme  34815  tgoldbachgtde  34817  tgoldbachgt  34820  kur14lem7  35406  brtpid3  35917  rabren3dioph  43057  oenord1ex  43557  fourierdlem114  46464  usgrexmpl1tri  48271  usgrexmpl2nb0  48277  usgrexmpl2nb1  48278  usgrexmpl2nb2  48279  usgrexmpl2nb3  48280  usgrexmpl2nb4  48281  usgrexmpl2nb5  48282  gpg3kgrtriex  48335