Theorem brtp 33717

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 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷) ∨ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹)))

Proof of Theorem brtp

Step	Hyp	Ref	Expression
1		df-br 5075	. 2 ⊢ (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩})
2		opex 5379	. . 3 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
3	2	eltp 4624	. 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
6	4, 5	opth 5391	. . 3 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵))
7	4, 5	opth 5391	. . 3 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷))
8	4, 5	opth 5391	. . 3 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝐸, 𝐹⟩ ↔ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹))
9	6, 7, 8	3orbi123i 1155	. 2 ⊢ ((⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩ ∨ ⟨𝑋, 𝑌⟩ = ⟨𝐶, 𝐷⟩ ∨ ⟨𝑋, 𝑌⟩ = ⟨𝐸, 𝐹⟩) ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷) ∨ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹)))
10	1, 3, 9	3bitri 297	1 ⊢ (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷) ∨ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹)))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∨ w3o 1085   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {ctp 4565  ⟨cop 4567   class class class wbr 5074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-br 5075

This theorem is referenced by:  sltval2  33859  sltintdifex  33864  sltres  33865  noextendlt  33872  noextendgt  33873  nolesgn2o  33874  nogesgn1o  33876  sltsolem1  33878  nosepnelem  33882  nosep1o  33884  nosep2o  33885  nosepdmlem  33886  nodenselem8  33894  nodense  33895  nolt02o  33898  nogt01o  33899  nosupbnd2lem1  33918  noinfbnd2lem1  33933