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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 6102 | . 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 35315 | . . 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 5776 | . 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 394 ∨ w3o 1084 ∧ w3a 1085 ∈ wcel 2104 ∃wrex 3068 ⟨cop 4633 class class class wbr 5147 ◡ccnv 5674 Rel wrel 5680 ‘cfv 6542 {coprab 7412 ℕcn 12216 𝔼cee 28413 Btwn cbtwn 28414 Colinear ccolin 35313 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 df-in 3954 df-ss 3964 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-colinear 35315 |
This theorem is referenced by: brcolinear2 35334 |
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