Theorem brprop 33209

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 4587	. . . 4 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
2	1	breqi 5109	. . 3 ⊢ (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ 𝑋({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})𝑌)
3		brun 5156	. . 3 ⊢ (𝑋({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})𝑌 ↔ (𝑋{⟨𝐴, 𝐵⟩}𝑌 ∨ 𝑋{⟨𝐶, 𝐷⟩}𝑌))
4	2, 3	bitri 278	. 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 5493	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)))
8	5, 6, 7	syl2anc 596	. . 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 5493	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (𝑋{⟨𝐶, 𝐷⟩}𝑌 ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)))
12	9, 10, 11	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑋{⟨𝐶, 𝐷⟩}𝑌 ↔ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷)))
13	8, 12	orbi12d 932	. 2 ⊢ (𝜑 → ((𝑋{⟨𝐴, 𝐵⟩}𝑌 ∨ 𝑋{⟨𝐶, 𝐷⟩}𝑌) ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷))))
14	4, 13	bitrid 286	1 ⊢ (𝜑 → (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ∪ cun 3897  {csn 4584  {cpr 4586  ⟨cop 4590   class class class wbr 5103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104

This theorem is used by: (None)