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Theorem brprop 31030
Description: Binary relation for a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.)
Hypotheses
Ref Expression
brprop.a (𝜑𝐴𝑉)
brprop.b (𝜑𝐵𝑊)
brprop.c (𝜑𝐶𝑉)
brprop.d (𝜑𝐷𝑊)
Assertion
Ref Expression
brprop (𝜑 → (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ ((𝑋 = 𝐴𝑌 = 𝐵) ∨ (𝑋 = 𝐶𝑌 = 𝐷))))

Proof of Theorem brprop
StepHypRef Expression
1 df-pr 4564 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
21breqi 5080 . . 3 (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌𝑋({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})𝑌)
3 brun 5125 . . 3 (𝑋({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})𝑌 ↔ (𝑋{⟨𝐴, 𝐵⟩}𝑌𝑋{⟨𝐶, 𝐷⟩}𝑌))
42, 3bitri 274 . 2 (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ (𝑋{⟨𝐴, 𝐵⟩}𝑌𝑋{⟨𝐶, 𝐷⟩}𝑌))
5 brprop.a . . . 4 (𝜑𝐴𝑉)
6 brprop.b . . . 4 (𝜑𝐵𝑊)
7 brsnop 5436 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴𝑌 = 𝐵)))
85, 6, 7syl2anc 584 . . 3 (𝜑 → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴𝑌 = 𝐵)))
9 brprop.c . . . 4 (𝜑𝐶𝑉)
10 brprop.d . . . 4 (𝜑𝐷𝑊)
11 brsnop 5436 . . . 4 ((𝐶𝑉𝐷𝑊) → (𝑋{⟨𝐶, 𝐷⟩}𝑌 ↔ (𝑋 = 𝐶𝑌 = 𝐷)))
129, 10, 11syl2anc 584 . . 3 (𝜑 → (𝑋{⟨𝐶, 𝐷⟩}𝑌 ↔ (𝑋 = 𝐶𝑌 = 𝐷)))
138, 12orbi12d 916 . 2 (𝜑 → ((𝑋{⟨𝐴, 𝐵⟩}𝑌𝑋{⟨𝐶, 𝐷⟩}𝑌) ↔ ((𝑋 = 𝐴𝑌 = 𝐵) ∨ (𝑋 = 𝐶𝑌 = 𝐷))))
144, 13syl5bb 283 1 (𝜑 → (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ ((𝑋 = 𝐴𝑌 = 𝐵) ∨ (𝑋 = 𝐶𝑌 = 𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844   = wceq 1539  wcel 2106  cun 3885  {csn 4561  {cpr 4563  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-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-op 4568  df-br 5075
This theorem is referenced by: (None)
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