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Theorem tpid2 3602
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
tpid2.1 𝐵 ∈ V
Assertion
Ref Expression
tpid2 𝐵 ∈ {𝐴, 𝐵, 𝐶}

Proof of Theorem tpid2
StepHypRef Expression
1 eqid 2115 . . 3 𝐵 = 𝐵
213mix2i 1137 . 2 (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)
3 tpid2.1 . . 3 𝐵 ∈ V
43eltp 3537 . 2 (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶))
52, 4mpbir 145 1 𝐵 ∈ {𝐴, 𝐵, 𝐶}
Colors of variables: wff set class
Syntax hints:  w3o 944   = wceq 1314  wcel 1463  Vcvv 2657  {ctp 3495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3or 946  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500  df-tp 3501
This theorem is referenced by: (None)
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