Theorem tpid1 4439

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 2771	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix1i 1417	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶)
3		tpid1.1	. . 3 ⊢ 𝐴 ∈ V
4	3	eltp 4367	. 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	2, 4	mpbir 221	1 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∨ w3o 1070   = wceq 1631   ∈ wcel 2145  Vcvv 3351  {ctp 4320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-un 3728  df-sn 4317  df-pr 4319  df-tp 4321

This theorem is referenced by:  tpnz  4447  wrdl3s3  13915  cffldtocusgr  26578  umgrwwlks2on  27105  sgnsf  30069  sgncl  30940  prodfzo03  31021  circlevma  31060  circlemethhgt  31061  hgt750lemg  31072  hgt750lemb  31074  hgt750lema  31075  hgt750leme  31076  tgoldbachgtde  31078  tgoldbachgt  31081  kur14lem7  31532  kur14lem9  31534  brtpid1  31940  rabren3dioph  37905  fourierdlem102  40942  fourierdlem114  40954  etransclem48  41016