Theorem brtp 5533

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 5149	. 2 ⊢ (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩})
2		opex 5475	. . 3 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
3	2	eltp 4694	. 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 5487	. . 3 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵))
7	4, 5	opth 5487	. . 3 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷))
8	4, 5	opth 5487	. . 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 206   ∧ wa 395   ∨ w3o 1085   = wceq 1537   ∈ wcel 2106  Vcvv 3478  {ctp 4635  ⟨cop 4637   class class class wbr 5148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-br 5149

This theorem is referenced by:  sltval2  27716  sltintdifex  27721  sltres  27722  noextendlt  27729  noextendgt  27730  nolesgn2o  27731  nogesgn1o  27733  sltsolem1  27735  nosepnelem  27739  nosep1o  27741  nosep2o  27742  nosepdmlem  27743  nodenselem8  27751  nodense  27752  nolt02o  27755  nogt01o  27756  nosupbnd2lem1  27775  noinfbnd2lem1  27790