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Theorem brtp 32882
Description: A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)
Hypotheses
Ref Expression
brtp.1 𝑋 ∈ V
brtp.2 𝑌 ∈ V
Assertion
Ref Expression
brtp (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩}𝑌 ↔ ((𝑋 = 𝐴𝑌 = 𝐵) ∨ (𝑋 = 𝐶𝑌 = 𝐷) ∨ (𝑋 = 𝐸𝑌 = 𝐹)))

Proof of Theorem brtp
StepHypRef Expression
1 df-br 5058 . 2 (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩})
2 opex 5347 . . 3 𝑋, 𝑌⟩ ∈ V
32eltp 4618 . 2 (⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} ↔ (⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩ ∨ ⟨𝑋, 𝑌⟩ = ⟨𝐶, 𝐷⟩ ∨ ⟨𝑋, 𝑌⟩ = ⟨𝐸, 𝐹⟩))
4 brtp.1 . . . 4 𝑋 ∈ V
5 brtp.2 . . . 4 𝑌 ∈ V
64, 5opth 5359 . . 3 (⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑋 = 𝐴𝑌 = 𝐵))
74, 5opth 5359 . . 3 (⟨𝑋, 𝑌⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑋 = 𝐶𝑌 = 𝐷))
84, 5opth 5359 . . 3 (⟨𝑋, 𝑌⟩ = ⟨𝐸, 𝐹⟩ ↔ (𝑋 = 𝐸𝑌 = 𝐹))
96, 7, 83orbi123i 1148 . 2 ((⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩ ∨ ⟨𝑋, 𝑌⟩ = ⟨𝐶, 𝐷⟩ ∨ ⟨𝑋, 𝑌⟩ = ⟨𝐸, 𝐹⟩) ↔ ((𝑋 = 𝐴𝑌 = 𝐵) ∨ (𝑋 = 𝐶𝑌 = 𝐷) ∨ (𝑋 = 𝐸𝑌 = 𝐹)))
101, 3, 93bitri 298 1 (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩}𝑌 ↔ ((𝑋 = 𝐴𝑌 = 𝐵) ∨ (𝑋 = 𝐶𝑌 = 𝐷) ∨ (𝑋 = 𝐸𝑌 = 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3o 1078   = wceq 1528  wcel 2105  Vcvv 3492  {ctp 4561  cop 4563   class class class wbr 5057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-br 5058
This theorem is referenced by:  sltval2  33060  sltintdifex  33065  sltres  33066  noextendlt  33073  noextendgt  33074  nolesgn2o  33075  sltsolem1  33077  nosepnelem  33081  nosep1o  33083  nosepdmlem  33084  nodenselem8  33092  nodense  33093  nolt02o  33096  nosupbnd2lem1  33112
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