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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 5064 | . 2 ⊢ (𝑋{〈𝐴, 𝐵〉}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉}) | |
2 | opex 5353 | . . . 4 ⊢ 〈𝑋, 𝑌〉 ∈ V | |
3 | 2 | elsn 4579 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑋, 𝑌〉 = 〈𝐴, 𝐵〉) |
4 | opthg2 5368 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝑋, 𝑌〉 = 〈𝐴, 𝐵〉 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵))) | |
5 | 3, 4 | syl5bb 285 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉} ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵))) |
6 | 1, 5 | syl5bb 285 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑋{〈𝐴, 𝐵〉}𝑌 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {csn 4564 〈cop 4570 class class class wbr 5063 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pr 5327 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3495 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-sn 4565 df-pr 4567 df-op 4571 df-br 5064 |
This theorem is referenced by: brprop 30433 |
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