Theorem brsnop 5505

Description: Binary relation for an ordered pair singleton. (Contributed by Thierry Arnoux, 23-Sep-2023.)

Assertion

Ref	Expression
brsnop	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)))

Proof of Theorem brsnop

Step	Hyp	Ref	Expression
1		df-br 5109	. 2 ⊢ (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩})
2		opex 5444	. . . 4 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
3	2	elsn 4603	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩)
4		opthg2 5460	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)))
5	3, 4	bitrid 286	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)))
6	1, 5	bitrid 286	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {csn 4588  ⟨cop 4594   class class class wbr 5108

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109

This theorem is used by:  brprop  33053  0funcg  49891  0funcALT  49894  functermc2  50315