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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 5067 | . 2 ⊢ (𝑋{〈𝐴, 𝐵〉}𝑌 ↔ 〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉}) | |
2 | opex 5356 | . . . 4 ⊢ 〈𝑋, 𝑌〉 ∈ V | |
3 | 2 | elsn 4582 | . . 3 ⊢ (〈𝑋, 𝑌〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑋, 𝑌〉 = 〈𝐴, 𝐵〉) |
4 | opthg2 5371 | . . 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 1537 ∈ wcel 2114 {csn 4567 〈cop 4573 class class class wbr 5066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 |
This theorem is referenced by: brprop 30433 |
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