brsegle - Mathbox for Scott Fenton

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Theorem brsegle 36109

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 5469	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		opex 5469	. . 3 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
3		eqeq1 2741	. . . . . . . 8 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩))
4		eqcom 2744	. . . . . . . 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 3179	. . . . 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 3222	. . . 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 3222	. . 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 2741	. . . . . . . 8 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (𝑞 = ⟨𝑐, 𝑑⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩))
11		eqcom 2744	. . . . . . . 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 3179	. . . . 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 3222	. . . 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 3222	. . 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 36108	. . 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 5548	. 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 3484	. . . . . . . . 9 ⊢ 𝑎 ∈ V
20		vex 3484	. . . . . . . . 9 ⊢ 𝑏 ∈ V
21	19, 20	opth 5481	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵))
22		vex 3484	. . . . . . . . 9 ⊢ 𝑐 ∈ V
23		vex 3484	. . . . . . . . 9 ⊢ 𝑑 ∈ V
24	22, 23	opth 5481	. . . . . . . 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 1156	. . . . . . 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 3129	. . . . . 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 3129	. . . . 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 3094	. . . 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 28911	. . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . 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 28928	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑛))) → 𝑁 = 𝑛)
38	33, 31, 34, 36, 37	syl22anc 839	. . . . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . . 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 3327	. . . . . . . . . . . . . 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 811	. . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
44		eleq1 2829	. . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝐶 → (𝑐 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑛)))
47		eleq1 2829	. . . . . . . . . . . . . . 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 4875	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
52	51	breq1d 5153	. . . . . . . . . . . . . . . 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 4875	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
55	54	breq2d 5155	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑦 Btwn ⟨𝑐, 𝑑⟩ ↔ 𝑦 Btwn ⟨𝐶, 𝐷⟩))
56		opeq1 4873	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝐶 → ⟨𝑐, 𝑦⟩ = ⟨𝐶, 𝑦⟩)
57	56	breq2d 5155	. . . . . . . . . . . . . . . . 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 3179	. . . . . . . . . . . . 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 3212	. . . . . 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 3213	. . . . 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 3155	. . . 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 2738	. . . . . . 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 2738	. . . . . . 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 4873	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝑑⟩)
81	80	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩))
82	80	breq2d 5155	. . . . . . . . . . 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 3179	. . . . . . . . 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 4874	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
87	86	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩))
88	86	breq2d 5155	. . . . . . . . . . 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 3179	. . . . . . . . 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 3635	. . . . . . 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 4873	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
95	94	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩))
96	94	breq1d 5153	. . . . . . . . . . 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 3179	. . . . . . . . 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 3222	. . . . . . 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 4874	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
102	101	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩))
103	101	breq1d 5153	. . . . . . . . . . 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 3179	. . . . . . . . 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 3222	. . . . . . 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 3635	. . . . . 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 6906	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
111	110	rexeqdv 3327	. . . . . . . . . . 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 3347	. . . . . . . . 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 3347	. . . . . . . 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 3347	. . . . . . 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 3347	. . . . . 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 3622	. . . . 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 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ⟨cop 4632 class class class wbr 5143 ‘cfv 6561 ℕcn 12266 𝔼cee 28903 Btwn cbtwn 28904 Cgrccgr 28905 Seg_≤ csegle 36107

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-z 12614 df-uz 12879 df-fz 13548 df-ee 28906 df-segle 36108

This theorem is referenced by: brsegle2 36110 seglecgr12im 36111 seglerflx 36113 seglemin 36114 segletr 36115 segleantisym 36116 seglelin 36117 btwnsegle 36118

W3C validator