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| Mirrors > Home > MPE Home > Th. List > Mathboxes > colinrel | Structured version Visualization version GIF version | ||
| Description: Colinearity is a relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| Ref | Expression |
|---|---|
| colinrel | ⊢ Rel Colinear |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | relcnv 6122 | . 2 ⊢ Rel ◡{〈〈𝑞, 𝑟〉, 𝑝〉 ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn 〈𝑞, 𝑟〉 ∨ 𝑞 Btwn 〈𝑟, 𝑝〉 ∨ 𝑟 Btwn 〈𝑝, 𝑞〉))} | |
| 2 | df-colinear 36040 | . . 3 ⊢ Colinear = ◡{〈〈𝑞, 𝑟〉, 𝑝〉 ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn 〈𝑞, 𝑟〉 ∨ 𝑞 Btwn 〈𝑟, 𝑝〉 ∨ 𝑟 Btwn 〈𝑝, 𝑞〉))} | |
| 3 | 2 | releqi 5787 | . 2 ⊢ (Rel Colinear ↔ Rel ◡{〈〈𝑞, 𝑟〉, 𝑝〉 ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn 〈𝑞, 𝑟〉 ∨ 𝑞 Btwn 〈𝑟, 𝑝〉 ∨ 𝑟 Btwn 〈𝑝, 𝑞〉))}) |
| 4 | 1, 3 | mpbir 231 | 1 ⊢ Rel Colinear |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∨ w3o 1086 ∧ w3a 1087 ∈ wcel 2108 ∃wrex 3070 〈cop 4632 class class class wbr 5143 ◡ccnv 5684 Rel wrel 5690 ‘cfv 6561 {coprab 7432 ℕcn 12266 𝔼cee 28903 Btwn cbtwn 28904 Colinear ccolin 36038 |
| 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-colinear 36040 |
| This theorem is referenced by: brcolinear2 36059 |
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