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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 5972 | . 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 34078 | . . 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 5649 | . 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 234 | 1 ⊢ Rel Colinear |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∨ w3o 1088 ∧ w3a 1089 ∈ wcel 2110 ∃wrex 3062 〈cop 4547 class class class wbr 5053 ◡ccnv 5550 Rel wrel 5556 ‘cfv 6380 {coprab 7214 ℕcn 11830 𝔼cee 26979 Btwn cbtwn 26980 Colinear ccolin 34076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-in 3873 df-ss 3883 df-opab 5116 df-xp 5557 df-rel 5558 df-cnv 5559 df-colinear 34078 |
This theorem is referenced by: brcolinear2 34097 |
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