Theorem brcolinear2 33632

Description: Alternate colinearity binary relation. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
brcolinear2	⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑃 Colinear ⟨𝑄, 𝑅⟩ ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑃⟩ ∨ 𝑅 Btwn ⟨𝑃, 𝑄⟩))))

Distinct variable groups:   𝑃,𝑛   𝑄,𝑛   𝑅,𝑛

Allowed substitution hints:   𝑉(𝑛)   𝑊(𝑛)

Proof of Theorem brcolinear2

Dummy variables 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		colinrel 33631	. . . 4 ⊢ Rel Colinear
2	1	brrelex1i 5572	. . 3 ⊢ (𝑃 Colinear ⟨𝑄, 𝑅⟩ → 𝑃 ∈ V)
3	2	a1i 11	. 2 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑃 Colinear ⟨𝑄, 𝑅⟩ → 𝑃 ∈ V))
4		elex 3459	. . . . . 6 ⊢ (𝑃 ∈ (𝔼‘𝑛) → 𝑃 ∈ V)
5	4	3ad2ant1 1130	. . . . 5 ⊢ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) → 𝑃 ∈ V)
6	5	adantr 484	. . . 4 ⊢ (((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑃⟩ ∨ 𝑅 Btwn ⟨𝑃, 𝑄⟩)) → 𝑃 ∈ V)
7	6	rexlimivw 3241	. . 3 ⊢ (∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑃⟩ ∨ 𝑅 Btwn ⟨𝑃, 𝑄⟩)) → 𝑃 ∈ V)
8	7	a1i 11	. 2 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑃⟩ ∨ 𝑅 Btwn ⟨𝑃, 𝑄⟩)) → 𝑃 ∈ V))
9		df-br 5031	. . . . . 6 ⊢ (𝑃 Colinear ⟨𝑄, 𝑅⟩ ↔ ⟨𝑃, ⟨𝑄, 𝑅⟩⟩ ∈ Colinear )
10		df-colinear 33613	. . . . . . 7 ⊢ 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⟩))}
11	10	eleq2i 2881	. . . . . 6 ⊢ (⟨𝑃, ⟨𝑄, 𝑅⟩⟩ ∈ Colinear ↔ ⟨𝑃, ⟨𝑄, 𝑅⟩⟩ ∈ ◡{⟨⟨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⟩))})
12	9, 11	bitri 278	. . . . 5 ⊢ (𝑃 Colinear ⟨𝑄, 𝑅⟩ ↔ ⟨𝑃, ⟨𝑄, 𝑅⟩⟩ ∈ ◡{⟨⟨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⟩))})
13		opex 5321	. . . . . . 7 ⊢ ⟨𝑄, 𝑅⟩ ∈ V
14		opelcnvg 5715	. . . . . . 7 ⊢ ((𝑃 ∈ V ∧ ⟨𝑄, 𝑅⟩ ∈ V) → (⟨𝑃, ⟨𝑄, 𝑅⟩⟩ ∈ ◡{⟨⟨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⟩))} ↔ ⟨⟨𝑄, 𝑅⟩, 𝑃⟩ ∈ {⟨⟨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⟩))}))
15	13, 14	mpan2 690	. . . . . 6 ⊢ (𝑃 ∈ V → (⟨𝑃, ⟨𝑄, 𝑅⟩⟩ ∈ ◡{⟨⟨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⟩))} ↔ ⟨⟨𝑄, 𝑅⟩, 𝑃⟩ ∈ {⟨⟨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⟩))}))
16	15	3ad2ant3 1132	. . . . 5 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑃 ∈ V) → (⟨𝑃, ⟨𝑄, 𝑅⟩⟩ ∈ ◡{⟨⟨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⟩))} ↔ ⟨⟨𝑄, 𝑅⟩, 𝑃⟩ ∈ {⟨⟨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⟩))}))
17	12, 16	syl5bb 286	. . . 4 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑃 ∈ V) → (𝑃 Colinear ⟨𝑄, 𝑅⟩ ↔ ⟨⟨𝑄, 𝑅⟩, 𝑃⟩ ∈ {⟨⟨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⟩))}))
18		eleq1 2877	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (𝑞 ∈ (𝔼‘𝑛) ↔ 𝑄 ∈ (𝔼‘𝑛)))
19	18	3anbi2d 1438	. . . . . . 7 ⊢ (𝑞 = 𝑄 → ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛)) ↔ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛))))
20		opeq1 4763	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → ⟨𝑞, 𝑟⟩ = ⟨𝑄, 𝑟⟩)
21	20	breq2d 5042	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (𝑝 Btwn ⟨𝑞, 𝑟⟩ ↔ 𝑝 Btwn ⟨𝑄, 𝑟⟩))
22		breq1 5033	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (𝑞 Btwn ⟨𝑟, 𝑝⟩ ↔ 𝑄 Btwn ⟨𝑟, 𝑝⟩))
23		opeq2 4765	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → ⟨𝑝, 𝑞⟩ = ⟨𝑝, 𝑄⟩)
24	23	breq2d 5042	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (𝑟 Btwn ⟨𝑝, 𝑞⟩ ↔ 𝑟 Btwn ⟨𝑝, 𝑄⟩))
25	21, 22, 24	3orbi123d 1432	. . . . . . 7 ⊢ (𝑞 = 𝑄 → ((𝑝 Btwn ⟨𝑞, 𝑟⟩ ∨ 𝑞 Btwn ⟨𝑟, 𝑝⟩ ∨ 𝑟 Btwn ⟨𝑝, 𝑞⟩) ↔ (𝑝 Btwn ⟨𝑄, 𝑟⟩ ∨ 𝑄 Btwn ⟨𝑟, 𝑝⟩ ∨ 𝑟 Btwn ⟨𝑝, 𝑄⟩)))
26	19, 25	anbi12d 633	. . . . . 6 ⊢ (𝑞 = 𝑄 → (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn ⟨𝑞, 𝑟⟩ ∨ 𝑞 Btwn ⟨𝑟, 𝑝⟩ ∨ 𝑟 Btwn ⟨𝑝, 𝑞⟩)) ↔ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn ⟨𝑄, 𝑟⟩ ∨ 𝑄 Btwn ⟨𝑟, 𝑝⟩ ∨ 𝑟 Btwn ⟨𝑝, 𝑄⟩))))
27	26	rexbidv 3256	. . . . 5 ⊢ (𝑞 = 𝑄 → (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn ⟨𝑞, 𝑟⟩ ∨ 𝑞 Btwn ⟨𝑟, 𝑝⟩ ∨ 𝑟 Btwn ⟨𝑝, 𝑞⟩)) ↔ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn ⟨𝑄, 𝑟⟩ ∨ 𝑄 Btwn ⟨𝑟, 𝑝⟩ ∨ 𝑟 Btwn ⟨𝑝, 𝑄⟩))))
28		eleq1 2877	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟 ∈ (𝔼‘𝑛) ↔ 𝑅 ∈ (𝔼‘𝑛)))
29	28	3anbi3d 1439	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛)) ↔ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛))))
30		opeq2 4765	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ⟨𝑄, 𝑟⟩ = ⟨𝑄, 𝑅⟩)
31	30	breq2d 5042	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑝 Btwn ⟨𝑄, 𝑟⟩ ↔ 𝑝 Btwn ⟨𝑄, 𝑅⟩))
32		opeq1 4763	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ⟨𝑟, 𝑝⟩ = ⟨𝑅, 𝑝⟩)
33	32	breq2d 5042	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑄 Btwn ⟨𝑟, 𝑝⟩ ↔ 𝑄 Btwn ⟨𝑅, 𝑝⟩))
34		breq1 5033	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟 Btwn ⟨𝑝, 𝑄⟩ ↔ 𝑅 Btwn ⟨𝑝, 𝑄⟩))
35	31, 33, 34	3orbi123d 1432	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑝 Btwn ⟨𝑄, 𝑟⟩ ∨ 𝑄 Btwn ⟨𝑟, 𝑝⟩ ∨ 𝑟 Btwn ⟨𝑝, 𝑄⟩) ↔ (𝑝 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑝⟩ ∨ 𝑅 Btwn ⟨𝑝, 𝑄⟩)))
36	29, 35	anbi12d 633	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn ⟨𝑄, 𝑟⟩ ∨ 𝑄 Btwn ⟨𝑟, 𝑝⟩ ∨ 𝑟 Btwn ⟨𝑝, 𝑄⟩)) ↔ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑝⟩ ∨ 𝑅 Btwn ⟨𝑝, 𝑄⟩))))
37	36	rexbidv 3256	. . . . 5 ⊢ (𝑟 = 𝑅 → (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn ⟨𝑄, 𝑟⟩ ∨ 𝑄 Btwn ⟨𝑟, 𝑝⟩ ∨ 𝑟 Btwn ⟨𝑝, 𝑄⟩)) ↔ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑝⟩ ∨ 𝑅 Btwn ⟨𝑝, 𝑄⟩))))
38		eleq1 2877	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ (𝔼‘𝑛) ↔ 𝑃 ∈ (𝔼‘𝑛)))
39	38	3anbi1d 1437	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ↔ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛))))
40		breq1 5033	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝 Btwn ⟨𝑄, 𝑅⟩ ↔ 𝑃 Btwn ⟨𝑄, 𝑅⟩))
41		opeq2 4765	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → ⟨𝑅, 𝑝⟩ = ⟨𝑅, 𝑃⟩)
42	41	breq2d 5042	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑄 Btwn ⟨𝑅, 𝑝⟩ ↔ 𝑄 Btwn ⟨𝑅, 𝑃⟩))
43		opeq1 4763	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → ⟨𝑝, 𝑄⟩ = ⟨𝑃, 𝑄⟩)
44	43	breq2d 5042	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑅 Btwn ⟨𝑝, 𝑄⟩ ↔ 𝑅 Btwn ⟨𝑃, 𝑄⟩))
45	40, 42, 44	3orbi123d 1432	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑝 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑝⟩ ∨ 𝑅 Btwn ⟨𝑝, 𝑄⟩) ↔ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑃⟩ ∨ 𝑅 Btwn ⟨𝑃, 𝑄⟩)))
46	39, 45	anbi12d 633	. . . . . 6 ⊢ (𝑝 = 𝑃 → (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑝⟩ ∨ 𝑅 Btwn ⟨𝑝, 𝑄⟩)) ↔ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑃⟩ ∨ 𝑅 Btwn ⟨𝑃, 𝑄⟩))))
47	46	rexbidv 3256	. . . . 5 ⊢ (𝑝 = 𝑃 → (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑝⟩ ∨ 𝑅 Btwn ⟨𝑝, 𝑄⟩)) ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑃⟩ ∨ 𝑅 Btwn ⟨𝑃, 𝑄⟩))))
48	27, 37, 47	eloprabg 7241	. . . 4 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑃 ∈ V) → (⟨⟨𝑄, 𝑅⟩, 𝑃⟩ ∈ {⟨⟨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⟩))} ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑃⟩ ∨ 𝑅 Btwn ⟨𝑃, 𝑄⟩))))
49	17, 48	bitrd 282	. . 3 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑃 ∈ V) → (𝑃 Colinear ⟨𝑄, 𝑅⟩ ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑃⟩ ∨ 𝑅 Btwn ⟨𝑃, 𝑄⟩))))
50	49	3expia 1118	. 2 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑃 ∈ V → (𝑃 Colinear ⟨𝑄, 𝑅⟩ ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑃⟩ ∨ 𝑅 Btwn ⟨𝑃, 𝑄⟩)))))
51	3, 8, 50	pm5.21ndd 384	1 ⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑃 Colinear ⟨𝑄, 𝑅⟩ ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑅 ∈ (𝔼‘𝑛)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∨ 𝑄 Btwn ⟨𝑅, 𝑃⟩ ∨ 𝑅 Btwn ⟨𝑃, 𝑄⟩))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441  ⟨cop 4531   class class class wbr 5030  ^◡ccnv 5518  ‘cfv 6324  {coprab 7136  ℕcn 11625  𝔼cee 26682   Btwn cbtwn 26683   Colinear ccolin 33611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-oprab 7139  df-colinear 33613

This theorem is referenced by:  brcolinear  33633