Theorem brprop 31713

Description: Binary relation for a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.)

Hypotheses

Ref	Expression
brprop.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
brprop.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
brprop.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
brprop.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)

Assertion

Ref	Expression
brprop	⊢ (𝜑 → (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷))))

Proof of Theorem brprop

Step	Hyp	Ref	Expression
1		df-pr 4609	. . . 4 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
2	1	breqi 5131	. . 3 ⊢ (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ 𝑋({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})𝑌)
3		brun 5176	. . 3 ⊢ (𝑋({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})𝑌 ↔ (𝑋{⟨𝐴, 𝐵⟩}𝑌 ∨ 𝑋{⟨𝐶, 𝐷⟩}𝑌))
4	2, 3	bitri 274	. 2 ⊢ (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ (𝑋{⟨𝐴, 𝐵⟩}𝑌 ∨ 𝑋{⟨𝐶, 𝐷⟩}𝑌))
5		brprop.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		brprop.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		brsnop 5499	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)))
8	5, 6, 7	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)))
9		brprop.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
10		brprop.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
11		brsnop 5499	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (𝑋{⟨𝐶, 𝐷⟩}𝑌 ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)))
12	9, 10, 11	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑋{⟨𝐶, 𝐷⟩}𝑌 ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)))
13	8, 12	orbi12d 917	. 2 ⊢ (𝜑 → ((𝑋{⟨𝐴, 𝐵⟩}𝑌 ∨ 𝑋{⟨𝐶, 𝐷⟩}𝑌) ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷))))
14	4, 13	bitrid 282	1 ⊢ (𝜑 → (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   ∪ cun 3926  {csn 4606  {cpr 4608  ⟨cop 4612   class class class wbr 5125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pr 5404

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3419  df-v 3461  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-br 5126

This theorem is referenced by: (None)