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Theorem tpid1 4278
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 2621 . . 3 𝐴 = 𝐴
213mix1i 1231 . 2 (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶)
3 tpid1.1 . . 3 𝐴 ∈ V
43eltp 4206 . 2 (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴𝐴 = 𝐵𝐴 = 𝐶))
52, 4mpbir 221 1 𝐴 ∈ {𝐴, 𝐵, 𝐶}
Colors of variables: wff setvar class
Syntax hints:  w3o 1035   = wceq 1480  wcel 1987  Vcvv 3189  {ctp 4157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-un 3564  df-sn 4154  df-pr 4156  df-tp 4158
This theorem is referenced by:  tpnz  4288  wrdl3s3  13646  cffldtocusgr  26243  umgrwwlks2on  26732  sgnsf  29532  sgncl  30399  kur14lem7  30929  kur14lem9  30931  brtpid1  31338  rabren3dioph  36886  fourierdlem102  39753  fourierdlem114  39765  etransclem48  39827
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