MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpid1 Structured version   Visualization version   GIF version

Theorem tpid1 4703
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
StepHypRef Expression
1 eqid 2826 . . 3 𝐴 = 𝐴
213mix1i 1327 . 2 (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶)
3 tpid1.1 . . 3 𝐴 ∈ V
43eltp 4625 . 2 (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶))
52, 4mpbir 232 1 𝐴 ∈ {𝐴, 𝐵, 𝐶}
Colors of variables: wff setvar class
Syntax hints:  w3o 1080   = wceq 1530  wcel 2107  Vcvv 3500  {ctp 4568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-v 3502  df-un 3945  df-sn 4565  df-pr 4567  df-tp 4569
This theorem is referenced by:  tpnz  4713  wrdl3s3  14321  cffldtocusgr  27162  umgrwwlks2on  27669  s3rn  30555  cyc3evpm  30725  sgnsf  30737  sgncl  31701  prodfzo03  31779  circlevma  31818  circlemethhgt  31819  hgt750lemg  31830  hgt750lemb  31832  hgt750lema  31833  hgt750leme  31834  tgoldbachgtde  31836  tgoldbachgt  31839  kur14lem7  32362  kur14lem9  32364  brtpid1  32854  rabren3dioph  39296  fourierdlem102  42378  fourierdlem114  42390  etransclem48  42452
  Copyright terms: Public domain W3C validator