Theorem brsnop 5464

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 5073	. 2 ⊢ (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩})
2		opex 5403	. . . 4 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
3	2	elsn 4570	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩)
4		opthg2 5419	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)))
5	3, 4	bitrid 284	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)))
6	1, 5	bitrid 284	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {csn 4555  ⟨cop 4561   class class class wbr 5072

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073

This theorem is referenced by:  brprop  32789  0funcg  49575  0funcALT  49578  functermc2  49999