Theorem tpid1 4716

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 2736	. . 3 ⊢ 𝐴 = 𝐴
2	1	3mix1i 1332	. 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶)
3		tpid1.1	. . 3 ⊢ 𝐴 ∈ V
4	3	eltp 4636	. 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	2, 4	mpbir 230	1 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∨ w3o 1085   = wceq 1540   ∈ wcel 2105  Vcvv 3441  {ctp 4577

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-un 3903  df-sn 4574  df-pr 4576  df-tp 4578

This theorem is referenced by:  tpnz  4727  wrdl3s3  14776  cffldtocusgr  28103  umgrwwlks2on  28610  s3rn  31507  cyc3evpm  31704  sgnsf  31716  sgncl  32805  prodfzo03  32883  circlevma  32922  circlemethhgt  32923  hgt750lemg  32934  hgt750lemb  32936  hgt750lema  32937  hgt750leme  32938  tgoldbachgtde  32940  tgoldbachgt  32943  kur14lem7  33473  kur14lem9  33475  brtpid1  33962  rabren3dioph  40907  fourierdlem102  44093  fourierdlem114  44105  etransclem48  44167