Theorem brtp 5490

Description: A necessary and sufficient condition for two sets to be related under a binary relation which is 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 5098	. 2 ⊢ (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩})
2		opex 5428	. . 3 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
3	2	eltp 4645	. 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 5441	. . 3 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵))
7	4, 5	opth 5441	. . 3 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷))
8	4, 5	opth 5441	. . 3 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝐸, 𝐹⟩ ↔ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹))
9	6, 7, 8	3orbi123i 1168	. 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 299	1 ⊢ (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷) ∨ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹)))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∨ w3o 1096   = wceq 1559   ∈ wcel 2141  Vcvv 3453  {ctp 4583  ⟨cop 4585   class class class wbr 5097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-br 5098

This theorem is referenced by:  ltsval2  27708  ltsintdifex  27713  ltsres  27714  noextendlt  27721  noextendgt  27722  nolesgn2o  27723  nogesgn1o  27725  ltssolem1  27727  nosepnelem  27731  nosep1o  27733  nosep2o  27734  nosepdmlem  27735  nodenselem8  27743  nodense  27744  nolt02o  27747  nogt01o  27748  nosupbnd2lem1  27767  noinfbnd2lem1  27782