Theorem tpid1 4721

Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Hypothesis

Ref	Expression
tpid1.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
tpid1	⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶}

Proof of Theorem tpid1

Step	Hyp	Ref	Expression
1		eqid 2731	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix1i 1334	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶)
3		tpid1.1	. . 3 ⊢ 𝐴 ∈ V
4	3	eltp 4642	. 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	2, 4	mpbir 231	1 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∨ w3o 1085   = wceq 1541   ∈ wcel 2111  Vcvv 3436  {ctp 4580

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 2113  ax-9 2121  ax-ext 2703

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 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3907  df-sn 4577  df-pr 4579  df-tp 4581

This theorem is referenced by:  tpnz  4732  hash3tpb  14402  wrdl3s3  14869  cffldtocusgr  29426  cffldtocusgrOLD  29427  umgrwwlks2on  29936  sgncl  32812  s3rnOLD  32925  cyc3evpm  33117  sgnsf  33129  prodfzo03  34614  circlevma  34653  circlemethhgt  34654  hgt750lemg  34665  hgt750lemb  34667  hgt750lema  34668  hgt750leme  34669  tgoldbachgtde  34671  tgoldbachgt  34674  kur14lem7  35254  kur14lem9  35256  brtpid1  35763  rabren3dioph  42854  fourierdlem102  46252  fourierdlem114  46264  etransclem48  46326  usgrexmpl1tri  48062  usgrexmpl2nb0  48068  usgrexmpl2nb1  48069  usgrexmpl2nb2  48070  usgrexmpl2nb3  48071  usgrexmpl2nb4  48072  usgrexmpl2nb5  48073  gpg3kgrtriex  48126