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Theorem tpid1 3472
 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
tpid1.1 A V
Assertion
Ref Expression
tpid1 A {A, B, 𝐶}

Proof of Theorem tpid1
StepHypRef Expression
1 eqid 2037 . . 3 A = A
213mix1i 1075 . 2 (A = A A = B A = 𝐶)
3 tpid1.1 . . 3 A V
43eltp 3409 . 2 (A {A, B, 𝐶} ↔ (A = A A = B A = 𝐶))
52, 4mpbir 134 1 A {A, B, 𝐶}
 Colors of variables: wff set class Syntax hints:   ∨ w3o 883   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {ctp 3369 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3or 885  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-tp 3375 This theorem is referenced by:  tpnz  3484
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