Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
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 5422 | . 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 31316 | . . 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 5125 | . 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 220 | 1 ⊢ Rel Colinear |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∨ w3o 1030 ∧ w3a 1031 ∈ wcel 1977 ∃wrex 2897 〈cop 4131 class class class wbr 4583 ◡ccnv 5037 Rel wrel 5043 ‘cfv 5804 {coprab 6550 ℕcn 10897 𝔼cee 25568 Btwn cbtwn 25569 Colinear ccolin 31314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-colinear 31316 |
This theorem is referenced by: brcolinear2 31335 |
Copyright terms: Public domain | W3C validator |