Theorem brsnop 5466

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 5075	. 2 ⊢ (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩})
2		opex 5405	. . . 4 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
3	2	elsn 4572	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩)
4		opthg2 5421	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)))
5	3, 4	bitrid 283	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)))
6	1, 5	bitrid 283	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {csn 4557  ⟨cop 4563   class class class wbr 5074

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075

This theorem is referenced by:  brprop  32758  0funcg  49548  0funcALT  49551  functermc2  49972