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Mirrors > Home > MPE Home > Th. List > brsnop | Structured version Visualization version GIF version |
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 5149 | . 2 ⊢ (𝑋{〈𝐴, 𝐵〉}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉}) | |
2 | opex 5475 | . . . 4 ⊢ 〈𝑋, 𝑌〉 ∈ V | |
3 | 2 | elsn 4646 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑋, 𝑌〉 = 〈𝐴, 𝐵〉) |
4 | opthg2 5490 | . . 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 1537 ∈ wcel 2106 {csn 4631 〈cop 4637 class class class wbr 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 |
This theorem is referenced by: brprop 32712 |
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