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Theorem tpid2 3717
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 2187 . . 3 𝐵 = 𝐵
213mix2i 1171 . 2 (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)
3 tpid2.1 . . 3 𝐵 ∈ V
43eltp 3652 . 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 978   = wceq 1363  wcel 2158  Vcvv 2749  {ctp 3606
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3or 980  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-tp 3612
This theorem is referenced by: (None)
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