brsegle - Mathbox for Scott Fenton

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Theorem brsegle 36096

Description: Binary relation form of the segment comparison relationship. (Contributed by Scott Fenton, 11-Oct-2013.)

**Assertion**
Ref	Expression
brsegle	⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩ Seg_≤ ⟨𝐶, 𝐷⟩ ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))

Distinct variable groups: 𝑦,𝐴 𝑦,𝑁 𝑦,𝐷 𝑦,𝐶 𝑦,𝐵

Proof of Theorem brsegle Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑛 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 5424	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		opex 5424	. . 3 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
3		eqeq1 2733	. . . . . . . 8 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩))
4		eqcom 2736	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
5	3, 4	bitrdi 287	. . . . . . 7 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩))
6	5	3anbi1d 1442	. . . . . 6 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
7	6	rexbidv 3157	. . . . 5 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
8	7	2rexbidv 3202	. . . 4 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
9	8	2rexbidv 3202	. . 3 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
10		eqeq1 2733	. . . . . . . 8 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (𝑞 = ⟨𝑐, 𝑑⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩))
11		eqcom 2736	. . . . . . . 8 ⊢ (⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ↔ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
12	10, 11	bitrdi 287	. . . . . . 7 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (𝑞 = ⟨𝑐, 𝑑⟩ ↔ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩))
13	12	3anbi2d 1443	. . . . . 6 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → ((⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
14	13	rexbidv 3157	. . . . 5 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
15	14	2rexbidv 3202	. . . 4 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
16	15	2rexbidv 3202	. . 3 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
17		df-segle 36095	. . 3 ⊢ Seg_≤ = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))}
18	1, 2, 9, 16, 17	brab 5503	. 2 ⊢ (⟨𝐴, 𝐵⟩ Seg_≤ ⟨𝐶, 𝐷⟩ ↔ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
19		vex 3451	. . . . . . . . 9 ⊢ 𝑎 ∈ V
20		vex 3451	. . . . . . . . 9 ⊢ 𝑏 ∈ V
21	19, 20	opth 5436	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵))
22		vex 3451	. . . . . . . . 9 ⊢ 𝑐 ∈ V
23		vex 3451	. . . . . . . . 9 ⊢ 𝑑 ∈ V
24	22, 23	opth 5436	. . . . . . . 8 ⊢ (⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷))
25		biid 261	. . . . . . . 8 ⊢ (∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))
26	21, 24, 25	3anbi123i 1155	. . . . . . 7 ⊢ ((⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
27	26	2rexbii 3109	. . . . . 6 ⊢ (∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
28	27	2rexbii 3109	. . . . 5 ⊢ (∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
29	28	rexbii 3076	. . . 4 ⊢ (∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
30		simpl2l 1227	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ (𝔼‘𝑁))
31	30	ad2antrl 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷))) ∧ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)) ∧ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛))))) → 𝐴 ∈ (𝔼‘𝑁))
32		eleenn 28823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
33	31, 32	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷))) ∧ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)) ∧ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛))))) → 𝑁 ∈ ℕ)
34		simprlr 779	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷))) ∧ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)) ∧ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛))))) → 𝑛 ∈ ℕ)
35		simprll 778	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)) ∧ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛)))) → 𝐴 ∈ (𝔼‘𝑛))
36	35	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷))) ∧ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)) ∧ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛))))) → 𝐴 ∈ (𝔼‘𝑛))
37		axdimuniq 28840	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑛))) → 𝑁 = 𝑛)
38	33, 31, 34, 36, 37	syl22anc 838	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷))) ∧ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)) ∧ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛))))) → 𝑁 = 𝑛)
39	38	fveq2d 6862	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷))) ∧ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)) ∧ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛))))) → (𝔼‘𝑁) = (𝔼‘𝑛))
40	39	rexeqdv 3300	. . . . . . . . . . . . . 14 ⊢ (((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷))) ∧ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)) ∧ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛))))) → (∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ↔ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
41	40	exbiri 810	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷))) → ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)) ∧ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛)))) → (∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))))
42	41	anassrs 467	. . . . . . . . . . . 12 ⊢ (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)) ∧ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛)))) → (∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))))
43		eleq1 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
44		eleq1 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
45	43, 44	bi2anan9 638	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛))))
46		eleq1 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝐶 → (𝑐 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑛)))
47		eleq1 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝐷 → (𝑑 ∈ (𝔼‘𝑛) ↔ 𝐷 ∈ (𝔼‘𝑛)))
48	46, 47	bi2anan9 638	. . . . . . . . . . . . . 14 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((𝑐 ∈ (𝔼‘𝑛) ∧ 𝑑 ∈ (𝔼‘𝑛)) ↔ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛))))
49	45, 48	bi2anan9 638	. . . . . . . . . . . . 13 ⊢ (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ (𝑐 ∈ (𝔼‘𝑛) ∧ 𝑑 ∈ (𝔼‘𝑛))) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)) ∧ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛)))))
50	49	anbi2d 630	. . . . . . . . . . . 12 ⊢ (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ (𝑐 ∈ (𝔼‘𝑛) ∧ 𝑑 ∈ (𝔼‘𝑛)))) ↔ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)) ∧ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛))))))
51		opeq12 4839	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
52	51	breq1d 5117	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩))
53	52	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩)))
54		opeq12 4839	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
55	54	breq2d 5119	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑦 Btwn ⟨𝑐, 𝑑⟩ ↔ 𝑦 Btwn ⟨𝐶, 𝐷⟩))
56		opeq1 4837	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝐶 → ⟨𝑐, 𝑦⟩ = ⟨𝐶, 𝑦⟩)
57	56	breq2d 5119	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝐶 → (⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))
58	57	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))
59	55, 58	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
60	53, 59	sylan9bb 509	. . . . . . . . . . . . . 14 ⊢ (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
61	60	rexbidv 3157	. . . . . . . . . . . . 13 ⊢ (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
62	61	imbi1d 341	. . . . . . . . . . . 12 ⊢ (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ((∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) ↔ (∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))))
63	42, 50, 62	3imtr4d 294	. . . . . . . . . . 11 ⊢ (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ (𝑐 ∈ (𝔼‘𝑛) ∧ 𝑑 ∈ (𝔼‘𝑛)))) → (∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))))
64	63	com12 32	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ (𝑐 ∈ (𝔼‘𝑛) ∧ 𝑑 ∈ (𝔼‘𝑛)))) → (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) → (∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))))
65	64	expd 415	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ (𝑐 ∈ (𝔼‘𝑛) ∧ 𝑑 ∈ (𝔼‘𝑛)))) → ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))))
66	65	3impd 1349	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ (𝑐 ∈ (𝔼‘𝑛) ∧ 𝑑 ∈ (𝔼‘𝑛)))) → (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
67	66	expr 456	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛))) → ((𝑐 ∈ (𝔼‘𝑛) ∧ 𝑑 ∈ (𝔼‘𝑛)) → (((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))))
68	67	rexlimdvv 3193	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛))) → (∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
69	68	rexlimdvva 3194	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) → (∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
70	69	rexlimdva 3134	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) ∧ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷) ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
71	29, 70	biimtrid 242	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
72		simpl1 1192	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → 𝑁 ∈ ℕ)
73		simpl2l 1227	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → 𝐴 ∈ (𝔼‘𝑁))
74		simpl2r 1228	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → 𝐵 ∈ (𝔼‘𝑁))
75		simpl3l 1229	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → 𝐶 ∈ (𝔼‘𝑁))
76		simpl3r 1230	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → 𝐷 ∈ (𝔼‘𝑁))
77		eqidd 2730	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
78		eqidd 2730	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
79		simpr 484	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))
80		opeq1 4837	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝑑⟩)
81	80	eqeq1d 2731	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩))
82	80	breq2d 5119	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝑦 Btwn ⟨𝑐, 𝑑⟩ ↔ 𝑦 Btwn ⟨𝐶, 𝑑⟩))
83	82, 57	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝐶, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
84	83	rexbidv 3157	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩) ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
85	81, 84	3anbi23d 1441	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → ((⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩)) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))))
86		opeq2 4838	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
87	86	eqeq1d 2731	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩))
88	86	breq2d 5119	. . . . . . . . . . 11 ⊢ (𝑑 = 𝐷 → (𝑦 Btwn ⟨𝐶, 𝑑⟩ ↔ 𝑦 Btwn ⟨𝐶, 𝐷⟩))
89	88	anbi1d 631	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ((𝑦 Btwn ⟨𝐶, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
90	89	rexbidv 3157	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
91	87, 90	3anbi23d 1441	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ((⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))))
92	85, 91	rspc2ev 3601	. . . . . . 7 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))) → ∃𝑐 ∈ (𝔼‘𝑁)∃𝑑 ∈ (𝔼‘𝑁)(⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩)))
93	75, 76, 77, 78, 79, 92	syl113anc 1384	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → ∃𝑐 ∈ (𝔼‘𝑁)∃𝑑 ∈ (𝔼‘𝑁)(⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩)))
94		opeq1 4837	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
95	94	eqeq1d 2731	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩))
96	94	breq1d 5117	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))
97	96	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
98	97	rexbidv 3157	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
99	95, 98	3anbi13d 1440	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ((⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ (⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
100	99	2rexbidv 3202	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∃𝑐 ∈ (𝔼‘𝑁)∃𝑑 ∈ (𝔼‘𝑁)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑐 ∈ (𝔼‘𝑁)∃𝑑 ∈ (𝔼‘𝑁)(⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
101		opeq2 4838	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
102	101	eqeq1d 2731	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩))
103	101	breq1d 5117	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩))
104	103	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩)))
105	104	rexbidv 3157	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩)))
106	102, 105	3anbi13d 1440	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩))))
107	106	2rexbidv 3202	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∃𝑐 ∈ (𝔼‘𝑁)∃𝑑 ∈ (𝔼‘𝑁)(⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑐 ∈ (𝔼‘𝑁)∃𝑑 ∈ (𝔼‘𝑁)(⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩))))
108	100, 107	rspc2ev 3601	. . . . . 6 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ ∃𝑐 ∈ (𝔼‘𝑁)∃𝑑 ∈ (𝔼‘𝑁)(⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩))) → ∃𝑎 ∈ (𝔼‘𝑁)∃𝑏 ∈ (𝔼‘𝑁)∃𝑐 ∈ (𝔼‘𝑁)∃𝑑 ∈ (𝔼‘𝑁)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
109	73, 74, 93, 108	syl3anc 1373	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → ∃𝑎 ∈ (𝔼‘𝑁)∃𝑏 ∈ (𝔼‘𝑁)∃𝑐 ∈ (𝔼‘𝑁)∃𝑑 ∈ (𝔼‘𝑁)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
110		fveq2 6858	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
111	110	rexeqdv 3300	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
112	111	3anbi3d 1444	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → ((⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
113	110, 112	rexeqbidv 3320	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑑 ∈ (𝔼‘𝑁)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
114	110, 113	rexeqbidv 3320	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑐 ∈ (𝔼‘𝑁)∃𝑑 ∈ (𝔼‘𝑁)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
115	110, 114	rexeqbidv 3320	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑏 ∈ (𝔼‘𝑁)∃𝑐 ∈ (𝔼‘𝑁)∃𝑑 ∈ (𝔼‘𝑁)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
116	110, 115	rexeqbidv 3320	. . . . . 6 ⊢ (𝑛 = 𝑁 → (∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑎 ∈ (𝔼‘𝑁)∃𝑏 ∈ (𝔼‘𝑁)∃𝑐 ∈ (𝔼‘𝑁)∃𝑑 ∈ (𝔼‘𝑁)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
117	116	rspcev 3588	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ∃𝑎 ∈ (𝔼‘𝑁)∃𝑏 ∈ (𝔼‘𝑁)∃𝑐 ∈ (𝔼‘𝑁)∃𝑑 ∈ (𝔼‘𝑁)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))) → ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
118	72, 109, 117	syl2anc 584	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)) → ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
119	118	ex 412	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) → ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))))
120	71, 119	impbid 212	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
121	18, 120	bitrid 283	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩ Seg_≤ ⟨𝐶, 𝐷⟩ ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃wrex 3053 ⟨cop 4595 class class class wbr 5107 ‘cfv 6511 ℕcn 12186 𝔼cee 28815 Btwn cbtwn 28816 Cgrccgr 28817 Seg_≤ csegle 36094

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-z 12530 df-uz 12794 df-fz 13469 df-ee 28818 df-segle 36095

This theorem is referenced by: brsegle2 36097 seglecgr12im 36098 seglerflx 36100 seglemin 36101 segletr 36102 segleantisym 36103 seglelin 36104 btwnsegle 36105

W3C validator