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Theorem tpid3 3514
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
tpid3.1 𝐶 ∈ V
Assertion
Ref Expression
tpid3 𝐶 ∈ {𝐴, 𝐵, 𝐶}

Proof of Theorem tpid3
StepHypRef Expression
1 eqid 2082 . . 3 𝐶 = 𝐶
213mix3i 1113 . 2 (𝐶 = 𝐴𝐶 = 𝐵𝐶 = 𝐶)
3 tpid3.1 . . 3 𝐶 ∈ V
43eltp 3448 . 2 (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐶 = 𝐴𝐶 = 𝐵𝐶 = 𝐶))
52, 4mpbir 144 1 𝐶 ∈ {𝐴, 𝐵, 𝐶}
Colors of variables: wff set class
Syntax hints:  w3o 919   = wceq 1285  wcel 1434  Vcvv 2602  {ctp 3408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3or 921  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-tp 3414
This theorem is referenced by: (None)
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