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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 6104 | . 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 35011 | . . 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 5778 | . 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 230 | 1 ⊢ Rel Colinear |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∨ w3o 1087 ∧ w3a 1088 ∈ wcel 2107 ∃wrex 3071 ⟨cop 4635 class class class wbr 5149 ◡ccnv 5676 Rel wrel 5682 ‘cfv 6544 {coprab 7410 ℕcn 12212 𝔼cee 28146 Btwn cbtwn 28147 Colinear ccolin 35009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-in 3956 df-ss 3966 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-colinear 35011 |
This theorem is referenced by: brcolinear2 35030 |
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