brsegle - Mathbox for Scott Fenton

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Theorem brsegle 35613

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,𝐵 𝑦,𝐶 𝑦,𝐷 𝑦,𝑁

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 5457	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		opex 5457	. . 3 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
3		eqeq1 2730	. . . . . . . 8 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩))
4		eqcom 2733	. . . . . . . 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 1436	. . . . . 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 3172	. . . . 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 3213	. . . 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 3213	. . 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 2730	. . . . . . . 8 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (𝑞 = ⟨𝑐, 𝑑⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩))
11		eqcom 2733	. . . . . . . 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 1437	. . . . . 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 3172	. . . . 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 3213	. . . 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 3213	. . 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 35612	. . 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 5536	. 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 3472	. . . . . . . . 9 ⊢ 𝑎 ∈ V
20		vex 3472	. . . . . . . . 9 ⊢ 𝑏 ∈ V
21	19, 20	opth 5469	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵))
22		vex 3472	. . . . . . . . 9 ⊢ 𝑐 ∈ V
23		vex 3472	. . . . . . . . 9 ⊢ 𝑑 ∈ V
24	22, 23	opth 5469	. . . . . . . 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 1152	. . . . . . 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 3123	. . . . . 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 3123	. . . . 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 3088	. . . 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 1223	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ (𝔼‘𝑁))
31	30	ad2antrl 725	. . . . . . . . . . . . . . . . . 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 28662	. . . . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . 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 28679	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑛))) → 𝑁 = 𝑛)
38	33, 31, 34, 36, 37	syl22anc 836	. . . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . 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 3320	. . . . . . . . . . . . . 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 808	. . . . . . . . . . . . 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 2815	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
44		eleq1 2815	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
45	43, 44	bi2anan9 636	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛))))
46		eleq1 2815	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝐶 → (𝑐 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑛)))
47		eleq1 2815	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝐷 → (𝑑 ∈ (𝔼‘𝑛) ↔ 𝐷 ∈ (𝔼‘𝑛)))
48	46, 47	bi2anan9 636	. . . . . . . . . . . . . 14 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((𝑐 ∈ (𝔼‘𝑛) ∧ 𝑑 ∈ (𝔼‘𝑛)) ↔ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛))))
49	45, 48	bi2anan9 636	. . . . . . . . . . . . 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 628	. . . . . . . . . . . 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 4870	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
52	51	breq1d 5151	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩))
53	52	anbi2d 628	. . . . . . . . . . . . . . 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 4870	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
55	54	breq2d 5153	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑦 Btwn ⟨𝑐, 𝑑⟩ ↔ 𝑦 Btwn ⟨𝐶, 𝐷⟩))
56		opeq1 4868	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝐶 → ⟨𝑐, 𝑦⟩ = ⟨𝐶, 𝑦⟩)
57	56	breq2d 5153	. . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . 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 3172	. . . . . . . . . . . . 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 1345	. . . . . . . 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 3204	. . . . . 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 3205	. . . . 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 3149	. . . 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 241	. . 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 1188	. . . . 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 1223	. . . . . 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 1224	. . . . . 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 1225	. . . . . . 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 1226	. . . . . . 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 2727	. . . . . . 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 2727	. . . . . . 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 4868	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝑑⟩)
81	80	eqeq1d 2728	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩))
82	80	breq2d 5153	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝑦 Btwn ⟨𝑐, 𝑑⟩ ↔ 𝑦 Btwn ⟨𝐶, 𝑑⟩))
83	82, 57	anbi12d 630	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝐶, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
84	83	rexbidv 3172	. . . . . . . . 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 1435	. . . . . . . 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 4869	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
87	86	eqeq1d 2728	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩))
88	86	breq2d 5153	. . . . . . . . . . 11 ⊢ (𝑑 = 𝐷 → (𝑦 Btwn ⟨𝐶, 𝑑⟩ ↔ 𝑦 Btwn ⟨𝐶, 𝐷⟩))
89	88	anbi1d 629	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → ((𝑦 Btwn ⟨𝐶, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
90	89	rexbidv 3172	. . . . . . . . 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 1435	. . . . . . . 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 3619	. . . . . . 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 1379	. . . . . 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 4868	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
95	94	eqeq1d 2728	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩))
96	94	breq1d 5151	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))
97	96	anbi2d 628	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
98	97	rexbidv 3172	. . . . . . . . 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 1434	. . . . . . . 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 3213	. . . . . . 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 4869	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
102	101	eqeq1d 2728	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩))
103	101	breq1d 5151	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩))
104	103	anbi2d 628	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩)))
105	104	rexbidv 3172	. . . . . . . . 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 1434	. . . . . . . 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 3213	. . . . . . 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 3619	. . . . . 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 1368	. . . . 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 6885	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
111	110	rexeqdv 3320	. . . . . . . . . . 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 1438	. . . . . . . . . 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 3337	. . . . . . . . 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 3337	. . . . . . . 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 3337	. . . . . . 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 3337	. . . . . 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 3606	. . . . 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 583	. . . 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 211	. 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 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 ⟨cop 4629 class class class wbr 5141 ‘cfv 6537 ℕcn 12216 𝔼cee 28654 Btwn cbtwn 28655 Cgrccgr 28656 Seg_≤ csegle 35611

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-z 12563 df-uz 12827 df-fz 13491 df-ee 28657 df-segle 35612

This theorem is referenced by: brsegle2 35614 seglecgr12im 35615 seglerflx 35617 seglemin 35618 segletr 35619 segleantisym 35620 seglelin 35621 btwnsegle 35622

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