Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brsegle Structured version   Visualization version   GIF version

Theorem brsegle 35737
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 5470 . . 3 𝐴, 𝐵⟩ ∈ V
2 opex 5470 . . 3 𝐶, 𝐷⟩ ∈ V
3 eqeq1 2732 . . . . . . . 8 (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩))
4 eqcom 2735 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
53, 4bitrdi 286 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩))
653anbi1d 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⟩))))
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 3217 . . . 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 3217 . . 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 2732 . . . . . . . 8 (𝑞 = ⟨𝐶, 𝐷⟩ → (𝑞 = ⟨𝑐, 𝑑⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩))
11 eqcom 2735 . . . . . . . 8 (⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ↔ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
1210, 11bitrdi 286 . . . . . . 7 (𝑞 = ⟨𝐶, 𝐷⟩ → (𝑞 = ⟨𝑐, 𝑑⟩ ↔ ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩))
13123anbi2d 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⟩))))
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 3217 . . . 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 3217 . . 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 35736 . . 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 5549 . 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 3477 . . . . . . . . 9 𝑎 ∈ V
20 vex 3477 . . . . . . . . 9 𝑏 ∈ V
2119, 20opth 5482 . . . . . . . 8 (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑎 = 𝐴𝑏 = 𝐵))
22 vex 3477 . . . . . . . . 9 𝑐 ∈ V
23 vex 3477 . . . . . . . . 9 𝑑 ∈ V
2422, 23opth 5482 . . . . . . . 8 (⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑐 = 𝐶𝑑 = 𝐷))
25 biid 260 . . . . . . . 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 1152 . . . . . . 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 1223 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ (𝔼‘𝑁))
3130ad2antrl 726 . . . . . . . . . . . . . . . . . 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 28727 . . . . . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . 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 28744 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑛))) → 𝑁 = 𝑛)
3833, 31, 34, 36, 37syl22anc 837 . . . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . 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 809 . . . . . . . . . . . . 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 466 . . . . . . . . . . . 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 2817 . . . . . . . . . . . . . . 15 (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
44 eleq1 2817 . . . . . . . . . . . . . . 15 (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
4543, 44bi2anan9 636 . . . . . . . . . . . . . 14 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛))))
46 eleq1 2817 . . . . . . . . . . . . . . 15 (𝑐 = 𝐶 → (𝑐 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑛)))
47 eleq1 2817 . . . . . . . . . . . . . . 15 (𝑑 = 𝐷 → (𝑑 ∈ (𝔼‘𝑛) ↔ 𝐷 ∈ (𝔼‘𝑛)))
4846, 47bi2anan9 636 . . . . . . . . . . . . . 14 ((𝑐 = 𝐶𝑑 = 𝐷) → ((𝑐 ∈ (𝔼‘𝑛) ∧ 𝑑 ∈ (𝔼‘𝑛)) ↔ (𝐶 ∈ (𝔼‘𝑛) ∧ 𝐷 ∈ (𝔼‘𝑛))))
4945, 48bi2anan9 636 . . . . . . . . . . . . 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 628 . . . . . . . . . . . 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 4880 . . . . . . . . . . . . . . . . 17 ((𝑎 = 𝐴𝑏 = 𝐵) → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
5251breq1d 5162 . . . . . . . . . . . . . . . 16 ((𝑎 = 𝐴𝑏 = 𝐵) → (⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩))
5352anbi2d 628 . . . . . . . . . . . . . . 15 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩)))
54 opeq12 4880 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝐶𝑑 = 𝐷) → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
5554breq2d 5164 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝐶𝑑 = 𝐷) → (𝑦 Btwn ⟨𝑐, 𝑑⟩ ↔ 𝑦 Btwn ⟨𝐶, 𝐷⟩))
56 opeq1 4878 . . . . . . . . . . . . . . . . . 18 (𝑐 = 𝐶 → ⟨𝑐, 𝑦⟩ = ⟨𝐶, 𝑦⟩)
5756breq2d 5164 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝐶 → (⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))
5857adantr 479 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝐶𝑑 = 𝐷) → (⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩))
5955, 58anbi12d 630 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑑 = 𝐷) → ((𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩) ↔ (𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
6053, 59sylan9bb 508 . . . . . . . . . . . . . 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 340 . . . . . . . . . . . 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 293 . . . . . . . . . . 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 414 . . . . . . . . 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 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 = 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 455 . . . . . . 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 3208 . . . . . 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 3209 . . . . 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 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 2729 . . . . . . 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 2729 . . . . . . 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 483 . . . . . . 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 4878 . . . . . . . . . 10 (𝑐 = 𝐶 → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝑑⟩)
8180eqeq1d 2730 . . . . . . . . 9 (𝑐 = 𝐶 → (⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩))
8280breq2d 5164 . . . . . . . . . . 11 (𝑐 = 𝐶 → (𝑦 Btwn ⟨𝑐, 𝑑⟩ ↔ 𝑦 Btwn ⟨𝐶, 𝑑⟩))
8382, 57anbi12d 630 . . . . . . . . . 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 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 4879 . . . . . . . . . 10 (𝑑 = 𝐷 → ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
8786eqeq1d 2730 . . . . . . . . 9 (𝑑 = 𝐷 → (⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩))
8886breq2d 5164 . . . . . . . . . . 11 (𝑑 = 𝐷 → (𝑦 Btwn ⟨𝐶, 𝑑⟩ ↔ 𝑦 Btwn ⟨𝐶, 𝐷⟩))
8988anbi1d 629 . . . . . . . . . 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 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⟩))))
9285, 91rspc2ev 3624 . . . . . . 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 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 4878 . . . . . . . . . 10 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
9594eqeq1d 2730 . . . . . . . . 9 (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩))
9694breq1d 5162 . . . . . . . . . . 11 (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))
9796anbi2d 628 . . . . . . . . . 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 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⟩))))
100992rexbidv 3217 . . . . . . 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 4879 . . . . . . . . . 10 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
102101eqeq1d 2730 . . . . . . . . 9 (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩))
103101breq1d 5162 . . . . . . . . . . 11 (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩Cgr⟨𝑐, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝑐, 𝑦⟩))
104103anbi2d 628 . . . . . . . . . 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 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⟩))))
1071062rexbidv 3217 . . . . . . 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 3624 . . . . . 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 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 6902 . . . . . . 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 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⟩))))
113110, 112rexeqbidv 3341 . . . . . . . . 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 3341 . . . . . . . 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 3341 . . . . . . 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 3341 . . . . . 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 3611 . . . . 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 582 . . . 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 411 . . 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 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⟩)))
12118, 120bitrid 282 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 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wrex 3067  cop 4638   class class class wbr 5152  cfv 6553  cn 12250  𝔼cee 28719   Btwn cbtwn 28720  Cgrccgr 28721   Seg csegle 35735
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-z 12597  df-uz 12861  df-fz 13525  df-ee 28722  df-segle 35736
This theorem is referenced by:  brsegle2  35738  seglecgr12im  35739  seglerflx  35741  seglemin  35742  segletr  35743  segleantisym  35744  seglelin  35745  btwnsegle  35746
  Copyright terms: Public domain W3C validator