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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 5953 | . 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 33507 | . . 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 5638 | . 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 233 | 1 ⊢ Rel Colinear |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∨ w3o 1082 ∧ w3a 1083 ∈ wcel 2114 ∃wrex 3139 〈cop 4559 class class class wbr 5052 ◡ccnv 5540 Rel wrel 5546 ‘cfv 6341 {coprab 7143 ℕcn 11624 𝔼cee 26660 Btwn cbtwn 26661 Colinear ccolin 33505 |
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 |
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 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-opab 5115 df-xp 5547 df-rel 5548 df-cnv 5549 df-colinear 33507 |
This theorem is referenced by: brcolinear2 33526 |
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