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Theorem tpid3 3509
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 2054 . . 3 𝐶 = 𝐶
213mix3i 1087 . 2 (𝐶 = 𝐴𝐶 = 𝐵𝐶 = 𝐶)
3 tpid3.1 . . 3 𝐶 ∈ V
43eltp 3443 . 2 (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐶 = 𝐴𝐶 = 𝐵𝐶 = 𝐶))
52, 4mpbir 138 1 𝐶 ∈ {𝐴, 𝐵, 𝐶}
Colors of variables: wff set class
Syntax hints:  w3o 893   = wceq 1257  wcel 1407  Vcvv 2572  {ctp 3402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-3or 895  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-un 2947  df-sn 3406  df-pr 3407  df-tp 3408
This theorem is referenced by: (None)
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