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Theorem brsegle 36089
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⟩ Seg𝐶, 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.
StepHypRef Expression
1 opex 5474 . . 3 𝐴, 𝐵⟩ ∈ V
2 opex 5474 . . 3 𝐶, 𝐷⟩ ∈ V
3 eqeq1 2738 . . . . . . . 8 (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩))
4 eqcom 2741 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
53, 4bitrdi 287 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩))
653anbi1d 1439 . . . . . 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⟩))))
76rexbidv 3176 . . . . 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⟩))))
872rexbidv 3219 . . . 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⟩))))
982rexbidv 3219 . . 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 2738 . . . . . . . 8 (𝑞 = ⟨𝐶, 𝐷⟩ → (𝑞 = ⟨𝑐, 𝑑⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩))
11 eqcom 2741 . . . . . . . 8 (⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ↔ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
1210, 11bitrdi 287 . . . . . . 7 (𝑞 = ⟨𝐶, 𝐷⟩ → (𝑞 = ⟨𝑐, 𝑑⟩ ↔ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩))
13123anbi2d 1440 . . . . . 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⟩))))
1413rexbidv 3176 . . . . 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⟩))))
15142rexbidv 3219 . . . 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⟩))))
16152rexbidv 3219 . . 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 36088 . . 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⟩))}
181, 2, 9, 16, 17brab 5552 . 2 (⟨𝐴, 𝐵⟩ Seg𝐶, 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 3481 . . . . . . . . 9 𝑎 ∈ V
20 vex 3481 . . . . . . . . 9 𝑏 ∈ V
2119, 20opth 5486 . . . . . . . 8 (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑎 = 𝐴𝑏 = 𝐵))
22 vex 3481 . . . . . . . . 9 𝑐 ∈ V
23 vex 3481 . . . . . . . . 9 𝑑 ∈ V
2422, 23opth 5486 . . . . . . . 8 (⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑐 = 𝐶𝑑 = 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⟩))
2621, 24, 253anbi123i 1154 . . . . . . 7 ((⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)) ↔ ((𝑎 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 𝐷) ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
27262rexbii 3126 . . . . . 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 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 𝐷) ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
28272rexbii 3126 . . . . 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 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 𝐷) ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
2928rexbii 3091 . . . 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 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 𝐷) ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
30 simpl2l 1225 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ (𝔼‘𝑁))
3130ad2antrl 728 . . . . . . . . . . . . . . . . . 18 (((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶𝑑 = 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 28925 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
3331, 32syl 17 . . . . . . . . . . . . . . . . 17 (((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶𝑑 = 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 = 𝐶𝑑 = 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))
3635adantl 481 . . . . . . . . . . . . . . . . 17 (((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶𝑑 = 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 28942 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑛))) → 𝑁 = 𝑛)
3833, 31, 34, 36, 37syl22anc 839 . . . . . . . . . . . . . . . 16 (((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶𝑑 = 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)
3938fveq2d 6910 . . . . . . . . . . . . . . 15 (((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶𝑑 = 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))
4039rexeqdv 3324 . . . . . . . . . . . . . 14 (((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶𝑑 = 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⟩)))
4140exbiri 811 . . . . . . . . . . . . 13 ((𝑎 = 𝐴 ∧ (𝑏 = 𝐵 ∧ (𝑐 = 𝐶𝑑 = 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⟩))))
4241anassrs 467 . . . . . . . . . . . 12 (((𝑎 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 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 2826 . . . . . . . . . . . . . . 15 (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
44 eleq1 2826 . . . . . . . . . . . . . . 15 (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
4543, 44bi2anan9 638 . . . . . . . . . . . . . 14 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛))))
46 eleq1 2826 . . . . . . . . . . . . . . 15 (𝑐 = 𝐶 → (𝑐 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑛)))
47 eleq1 2826 . . . . . . . . . . . . . . 15 (𝑑 = 𝐷 → (𝑑 ∈ (𝔼‘𝑛) ↔ 𝐷 ∈ (𝔼‘𝑛)))
4846, 47bi2anan9 638 . . . . . . . . . . . . . 14 ((𝑐 = 𝐶𝑑 = 𝐷) → ((𝑐 ∈ (𝔼‘𝑛) ∧ 𝑑 ∈ (𝔼‘𝑛)) ↔ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛))))
4945, 48bi2anan9 638 . . . . . . . . . . . . 13 (((𝑎 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 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)))))
5049anbi2d 630 . . . . . . . . . . . 12 (((𝑎 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 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 4879 . . . . . . . . . . . . . . . . 17 ((𝑎 = 𝐴𝑏 = 𝐵) → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
5251breq1d 5157 . . . . . . . . . . . . . . . 16 ((𝑎 = 𝐴𝑏 = 𝐵) → (⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩))
5352anbi2d 630 . . . . . . . . . . . . . . 15 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩)))
54 opeq12 4879 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑑 = 𝐷) → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
5554breq2d 5159 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝐶𝑑 = 𝐷) → (𝑦 Btwn ⟨𝑐, 𝑑⟩ ↔ 𝑦 Btwn ⟨𝐶, 𝐷⟩))
56 opeq1 4877 . . . . . . . . . . . . . . . . . 18 (𝑐 = 𝐶 → ⟨𝑐, 𝑦⟩ = ⟨𝐶, 𝑦⟩)
5756breq2d 5159 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝐶 → (⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))
5857adantr 480 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝐶𝑑 = 𝐷) → (⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))
5955, 58anbi12d 632 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑑 = 𝐷) → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
6053, 59sylan9bb 509 . . . . . . . . . . . . . 14 (((𝑎 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 𝐷)) → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
6160rexbidv 3176 . . . . . . . . . . . . 13 (((𝑎 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 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⟩)))
6261imbi1d 341 . . . . . . . . . . . 12 (((𝑎 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 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⟩))))
6342, 50, 623imtr4d 294 . . . . . . . . . . 11 (((𝑎 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 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⟩))))
6463com12 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 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 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⟩))))
6564expd 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 = 𝐴𝑏 = 𝐵) → ((𝑐 = 𝐶𝑑 = 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⟩)))))
66653impd 1347 . . . . . . . 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 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 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⟩)))
6766expr 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 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 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⟩))))
6867rexlimdvv 3209 . . . . . 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 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 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⟩)))
6968rexlimdvva 3210 . . . . 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 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 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⟩)))
7069rexlimdva 3152 . . . 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 = 𝐴𝑏 = 𝐵) ∧ (𝑐 = 𝐶𝑑 = 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⟩)))
7129, 70biimtrid 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 1190 . . . . 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 1225 . . . . . 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 1226 . . . . . 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 1227 . . . . . . 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 1228 . . . . . . 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 2735 . . . . . . 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 2735 . . . . . . 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 4877 . . . . . . . . . 10 (𝑐 = 𝐶 → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝑑⟩)
8180eqeq1d 2736 . . . . . . . . 9 (𝑐 = 𝐶 → (⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩))
8280breq2d 5159 . . . . . . . . . . 11 (𝑐 = 𝐶 → (𝑦 Btwn ⟨𝑐, 𝑑⟩ ↔ 𝑦 Btwn ⟨𝐶, 𝑑⟩))
8382, 57anbi12d 632 . . . . . . . . . 10 (𝑐 = 𝐶 → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝐶, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
8483rexbidv 3176 . . . . . . . . 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⟩)))
8581, 843anbi23d 1438 . . . . . . . 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 4878 . . . . . . . . . 10 (𝑑 = 𝐷 → ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
8786eqeq1d 2736 . . . . . . . . 9 (𝑑 = 𝐷 → (⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩))
8886breq2d 5159 . . . . . . . . . . 11 (𝑑 = 𝐷 → (𝑦 Btwn ⟨𝐶, 𝑑⟩ ↔ 𝑦 Btwn ⟨𝐶, 𝐷⟩))
8988anbi1d 631 . . . . . . . . . 10 (𝑑 = 𝐷 → ((𝑦 Btwn ⟨𝐶, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
9089rexbidv 3176 . . . . . . . . 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⟩)))
9187, 903anbi23d 1438 . . . . . . . 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⟩))))
9285, 91rspc2ev 3634 . . . . . . 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⟩)))
9375, 76, 77, 78, 79, 92syl113anc 1381 . . . . . 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 4877 . . . . . . . . . 10 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
9594eqeq1d 2736 . . . . . . . . 9 (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩))
9694breq1d 5157 . . . . . . . . . . 11 (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))
9796anbi2d 630 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)))
9897rexbidv 3176 . . . . . . . . 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⟩)))
9995, 983anbi13d 1437 . . . . . . . 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⟩))))
100992rexbidv 3219 . . . . . . 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 4878 . . . . . . . . . 10 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
102101eqeq1d 2736 . . . . . . . . 9 (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩))
103101breq1d 5157 . . . . . . . . . . 11 (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩))
104103anbi2d 630 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩)))
105104rexbidv 3176 . . . . . . . . 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⟩)))
106102, 1053anbi13d 1437 . . . . . . . 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⟩))))
1071062rexbidv 3219 . . . . . . 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⟩))))
108100, 107rspc2ev 3634 . . . . . 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⟩)))
10973, 74, 93, 108syl3anc 1370 . . . . 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))
111110rexeqdv 3324 . . . . . . . . . . 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⟩)))
1121113anbi3d 1441 . . . . . . . . . 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⟩))))
113110, 112rexeqbidv 3344 . . . . . . . . 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⟩))))
114110, 113rexeqbidv 3344 . . . . . . . 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⟩))))
115110, 114rexeqbidv 3344 . . . . . . 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⟩))))
116110, 115rexeqbidv 3344 . . . . . 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⟩))))
117116rspcev 3621 . . . . 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⟩)))
11872, 109, 117syl2anc 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⟩)))
119118ex 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⟩))))
12071, 119impbid 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⟩)))
12118, 120bitrid 283 1 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩ Seg𝐶, 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 1536  wcel 2105  wrex 3067  cop 4636   class class class wbr 5147  cfv 6562  cn 12263  𝔼cee 28917   Btwn cbtwn 28918  Cgrccgr 28919   Seg csegle 36087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-z 12611  df-uz 12876  df-fz 13544  df-ee 28920  df-segle 36088
This theorem is referenced by:  brsegle2  36090  seglecgr12im  36091  seglerflx  36093  seglemin  36094  segletr  36095  segleantisym  36096  seglelin  36097  btwnsegle  36098
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