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Theorem tpid2 3705
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 2177 . . 3 𝐵 = 𝐵
213mix2i 1170 . 2 (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)
3 tpid2.1 . . 3 𝐵 ∈ V
43eltp 3640 . 2 (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶))
52, 4mpbir 146 1 𝐵 ∈ {𝐴, 𝐵, 𝐶}
Colors of variables: wff set class
Syntax hints:  w3o 977   = wceq 1353  wcel 2148  Vcvv 2737  {ctp 3594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3or 979  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-tp 3600
This theorem is referenced by: (None)
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