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Theorem tpid2 3482
 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 2040 . . 3 𝐵 = 𝐵
213mix2i 1077 . 2 (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶)
3 tpid2.1 . . 3 𝐵 ∈ V
43eltp 3418 . 2 (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴𝐵 = 𝐵𝐵 = 𝐶))
52, 4mpbir 134 1 𝐵 ∈ {𝐴, 𝐵, 𝐶}
 Colors of variables: wff set class Syntax hints:   ∨ w3o 884   = wceq 1243   ∈ wcel 1393  Vcvv 2557  {ctp 3377 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3or 886  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-tp 3383 This theorem is referenced by: (None)
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