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| Description: Binary relation for an ordered pair singleton. (Contributed by Thierry Arnoux, 23-Sep-2023.) | 
| Ref | Expression | 
|---|---|
| brsnop | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑋{〈𝐴, 𝐵〉}𝑌 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-br 5144 | . 2 ⊢ (𝑋{〈𝐴, 𝐵〉}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉}) | |
| 2 | opex 5469 | . . . 4 ⊢ 〈𝑋, 𝑌〉 ∈ V | |
| 3 | 2 | elsn 4641 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑋, 𝑌〉 = 〈𝐴, 𝐵〉) | 
| 4 | opthg2 5484 | . . 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))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 〈cop 4632 class class class wbr 5143 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 | 
| This theorem is referenced by: brprop 32706 0funcg 48918 0funcALT 48921 functermc2 49141 | 
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