Step | Hyp | Ref
| Expression |
1 | | ccolin 35009 |
. 2
class
Colinear |
2 | | va |
. . . . . . . . 9
setvar 𝑎 |
3 | 2 | cv 1541 |
. . . . . . . 8
class 𝑎 |
4 | | vn |
. . . . . . . . . 10
setvar 𝑛 |
5 | 4 | cv 1541 |
. . . . . . . . 9
class 𝑛 |
6 | | cee 28146 |
. . . . . . . . 9
class
𝔼 |
7 | 5, 6 | cfv 6544 |
. . . . . . . 8
class
(𝔼‘𝑛) |
8 | 3, 7 | wcel 2107 |
. . . . . . 7
wff 𝑎 ∈ (𝔼‘𝑛) |
9 | | vb |
. . . . . . . . 9
setvar 𝑏 |
10 | 9 | cv 1541 |
. . . . . . . 8
class 𝑏 |
11 | 10, 7 | wcel 2107 |
. . . . . . 7
wff 𝑏 ∈ (𝔼‘𝑛) |
12 | | vc |
. . . . . . . . 9
setvar 𝑐 |
13 | 12 | cv 1541 |
. . . . . . . 8
class 𝑐 |
14 | 13, 7 | wcel 2107 |
. . . . . . 7
wff 𝑐 ∈ (𝔼‘𝑛) |
15 | 8, 11, 14 | w3a 1088 |
. . . . . 6
wff (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) |
16 | 10, 13 | cop 4635 |
. . . . . . . 8
class
⟨𝑏, 𝑐⟩ |
17 | | cbtwn 28147 |
. . . . . . . 8
class
Btwn |
18 | 3, 16, 17 | wbr 5149 |
. . . . . . 7
wff 𝑎 Btwn ⟨𝑏, 𝑐⟩ |
19 | 13, 3 | cop 4635 |
. . . . . . . 8
class
⟨𝑐, 𝑎⟩ |
20 | 10, 19, 17 | wbr 5149 |
. . . . . . 7
wff 𝑏 Btwn ⟨𝑐, 𝑎⟩ |
21 | 3, 10 | cop 4635 |
. . . . . . . 8
class
⟨𝑎, 𝑏⟩ |
22 | 13, 21, 17 | wbr 5149 |
. . . . . . 7
wff 𝑐 Btwn ⟨𝑎, 𝑏⟩ |
23 | 18, 20, 22 | w3o 1087 |
. . . . . 6
wff (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩) |
24 | 15, 23 | wa 397 |
. . . . 5
wff ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩)) |
25 | | cn 12212 |
. . . . 5
class
ℕ |
26 | 24, 4, 25 | wrex 3071 |
. . . 4
wff
∃𝑛 ∈
ℕ ((𝑎 ∈
(𝔼‘𝑛) ∧
𝑏 ∈
(𝔼‘𝑛) ∧
𝑐 ∈
(𝔼‘𝑛)) ∧
(𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩)) |
27 | 26, 9, 12, 2 | coprab 7410 |
. . 3
class
{⟨⟨𝑏,
𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))} |
28 | 27 | ccnv 5676 |
. 2
class ◡{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))} |
29 | 1, 28 | wceq 1542 |
1
wff Colinear =
◡{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))} |