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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 5148 | . 2 ⊢ (𝑋{〈𝐴, 𝐵〉}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉}) | |
2 | opex 5463 | . . . 4 ⊢ 〈𝑋, 𝑌〉 ∈ V | |
3 | 2 | elsn 4642 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑋, 𝑌〉 = 〈𝐴, 𝐵〉) |
4 | opthg2 5478 | . . 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 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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