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Theorem brprop 30607
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 4519 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
21breqi 5036 . . 3 (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌𝑋({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})𝑌)
3 brun 5081 . . 3 (𝑋({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})𝑌 ↔ (𝑋{⟨𝐴, 𝐵⟩}𝑌𝑋{⟨𝐶, 𝐷⟩}𝑌))
42, 3bitri 278 . 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 5379 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴𝑌 = 𝐵)))
85, 6, 7syl2anc 587 . . 3 (𝜑 → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴𝑌 = 𝐵)))
9 brprop.c . . . 4 (𝜑𝐶𝑉)
10 brprop.d . . . 4 (𝜑𝐷𝑊)
11 brsnop 5379 . . . 4 ((𝐶𝑉𝐷𝑊) → (𝑋{⟨𝐶, 𝐷⟩}𝑌 ↔ (𝑋 = 𝐶𝑌 = 𝐷)))
129, 10, 11syl2anc 587 . . 3 (𝜑 → (𝑋{⟨𝐶, 𝐷⟩}𝑌 ↔ (𝑋 = 𝐶𝑌 = 𝐷)))
138, 12orbi12d 918 . 2 (𝜑 → ((𝑋{⟨𝐴, 𝐵⟩}𝑌𝑋{⟨𝐶, 𝐷⟩}𝑌) ↔ ((𝑋 = 𝐴𝑌 = 𝐵) ∨ (𝑋 = 𝐶𝑌 = 𝐷))))
144, 13syl5bb 286 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 209  wa 399  wo 846   = wceq 1542  wcel 2114  cun 3841  {csn 4516  {cpr 4518  cop 4522   class class class wbr 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-dif 3846  df-un 3848  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031
This theorem is referenced by: (None)
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