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Theorem brtp 32888
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 5064 . 2 (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩})
2 opex 5353 . . 3 𝑋, 𝑌⟩ ∈ V
32eltp 4625 . 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 5365 . . 3 (⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑋 = 𝐴𝑌 = 𝐵))
74, 5opth 5365 . . 3 (⟨𝑋, 𝑌⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑋 = 𝐶𝑌 = 𝐷))
84, 5opth 5365 . . 3 (⟨𝑋, 𝑌⟩ = ⟨𝐸, 𝐹⟩ ↔ (𝑋 = 𝐸𝑌 = 𝐹))
96, 7, 83orbi123i 1150 . 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 1080   = wceq 1530  wcel 2107  Vcvv 3500  {ctp 4568  cop 4570   class class class wbr 5063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-br 5064
This theorem is referenced by:  sltval2  33066  sltintdifex  33071  sltres  33072  noextendlt  33079  noextendgt  33080  nolesgn2o  33081  sltsolem1  33083  nosepnelem  33087  nosep1o  33089  nosepdmlem  33090  nodenselem8  33098  nodense  33099  nolt02o  33102  nosupbnd2lem1  33118
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