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Theorem brsnop 5521
Description: Binary relation for an ordered pair singleton. (Contributed by Thierry Arnoux, 23-Sep-2023.)
Assertion
Ref Expression
brsnop ((𝐴𝑉𝐵𝑊) → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴𝑌 = 𝐵)))

Proof of Theorem brsnop
StepHypRef Expression
1 df-br 5148 . 2 (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩})
2 opex 5463 . . . 4 𝑋, 𝑌⟩ ∈ V
32elsn 4642 . . 3 (⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩)
4 opthg2 5478 . . 3 ((𝐴𝑉𝐵𝑊) → (⟨𝑋, 𝑌⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑋 = 𝐴𝑌 = 𝐵)))
53, 4bitrid 283 . 2 ((𝐴𝑉𝐵𝑊) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ (𝑋 = 𝐴𝑌 = 𝐵)))
61, 5bitrid 283 1 ((𝐴𝑉𝐵𝑊) → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴𝑌 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {csn 4627  cop 4633   class class class wbr 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148
This theorem is referenced by:  brprop  31897
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