Theorem btwnconn1lem2 36069

Description: Lemma for btwnconn1 36082. Now, we show that two of the hypotheticals we introduced in the first lemma are identical. (Contributed by Scott Fenton, 8-Oct-2013.)

Assertion

Ref	Expression
btwnconn1lem2	⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → 𝑋 = 𝑏)

Proof of Theorem btwnconn1lem2

Step	Hyp	Ref	Expression
1		simp1ll 1237	. . 3 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩))) → 𝐴 ≠ 𝐵)
2	1	adantl 481	. 2 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → 𝐴 ≠ 𝐵)
3		simp11 1204	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
4		simp12 1205	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
5		simp13 1206	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
6		simp21 1207	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁))
7		simp33 1212	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) → 𝑋 ∈ (𝔼‘𝑁))
8		simp1rl 1239	. . . . 5 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩))) → 𝐵 Btwn ⟨𝐴, 𝐶⟩)
9	8	adantl 481	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → 𝐵 Btwn ⟨𝐴, 𝐶⟩)
10		simp2rl 1243	. . . . . . 7 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩))) → 𝐶 Btwn ⟨𝐴, 𝑑⟩)
11		simp3rl 1247	. . . . . . 7 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩))) → 𝑑 Btwn ⟨𝐴, 𝑋⟩)
12	10, 11	jca 511	. . . . . 6 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩))) → (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ 𝑑 Btwn ⟨𝐴, 𝑋⟩))
13	12	adantl 481	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ 𝑑 Btwn ⟨𝐴, 𝑋⟩))
14		simp31 1210	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) → 𝑑 ∈ (𝔼‘𝑁))
15		btwnexch 36006	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) → ((𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ 𝑑 Btwn ⟨𝐴, 𝑋⟩) → 𝐶 Btwn ⟨𝐴, 𝑋⟩))
16	3, 4, 6, 14, 7, 15	syl122anc 1381	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) → ((𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ 𝑑 Btwn ⟨𝐴, 𝑋⟩) → 𝐶 Btwn ⟨𝐴, 𝑋⟩))
17	16	adantr 480	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → ((𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ 𝑑 Btwn ⟨𝐴, 𝑋⟩) → 𝐶 Btwn ⟨𝐴, 𝑋⟩))
18	13, 17	mpd 15	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → 𝐶 Btwn ⟨𝐴, 𝑋⟩)
19	3, 4, 5, 6, 7, 9, 18	btwnexchand 36007	. . 3 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → 𝐵 Btwn ⟨𝐴, 𝑋⟩)
20	3, 5, 7	cgrrflx2d 35965	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) → ⟨𝐵, 𝑋⟩Cgr⟨𝑋, 𝐵⟩)
21	20	adantr 480	. . 3 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → ⟨𝐵, 𝑋⟩Cgr⟨𝑋, 𝐵⟩)
22	19, 21	jca 511	. 2 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → (𝐵 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝐵, 𝑋⟩Cgr⟨𝑋, 𝐵⟩))
23		simp23 1209	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) → 𝑐 ∈ (𝔼‘𝑁))
24		simp32 1211	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) → 𝑏 ∈ (𝔼‘𝑁))
25		simp1rr 1240	. . . . . . 7 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩))) → 𝐵 Btwn ⟨𝐴, 𝐷⟩)
26		simp2ll 1241	. . . . . . 7 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩))) → 𝐷 Btwn ⟨𝐴, 𝑐⟩)
27	25, 26	jca 511	. . . . . 6 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩))) → (𝐵 Btwn ⟨𝐴, 𝐷⟩ ∧ 𝐷 Btwn ⟨𝐴, 𝑐⟩))
28	27	adantl 481	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → (𝐵 Btwn ⟨𝐴, 𝐷⟩ ∧ 𝐷 Btwn ⟨𝐴, 𝑐⟩))
29		simp22 1208	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) → 𝐷 ∈ (𝔼‘𝑁))
30		btwnexch 36006	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝐴, 𝐷⟩ ∧ 𝐷 Btwn ⟨𝐴, 𝑐⟩) → 𝐵 Btwn ⟨𝐴, 𝑐⟩))
31	3, 4, 5, 29, 23, 30	syl122anc 1381	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝐴, 𝐷⟩ ∧ 𝐷 Btwn ⟨𝐴, 𝑐⟩) → 𝐵 Btwn ⟨𝐴, 𝑐⟩))
32	31	adantr 480	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → ((𝐵 Btwn ⟨𝐴, 𝐷⟩ ∧ 𝐷 Btwn ⟨𝐴, 𝑐⟩) → 𝐵 Btwn ⟨𝐴, 𝑐⟩))
33	28, 32	mpd 15	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → 𝐵 Btwn ⟨𝐴, 𝑐⟩)
34		simp3ll 1245	. . . . 5 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩))) → 𝑐 Btwn ⟨𝐴, 𝑏⟩)
35	34	adantl 481	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → 𝑐 Btwn ⟨𝐴, 𝑏⟩)
36	3, 4, 5, 23, 24, 33, 35	btwnexchand 36007	. . 3 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → 𝐵 Btwn ⟨𝐴, 𝑏⟩)
37	3, 4, 5, 23, 24, 33, 35	btwnexch3and 36002	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → 𝑐 Btwn ⟨𝐵, 𝑏⟩)
38	3, 4, 5, 6, 7, 9, 18	btwnexch3and 36002	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → 𝐶 Btwn ⟨𝐵, 𝑋⟩)
39	3, 6, 5, 7, 38	btwncomand 35996	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → 𝐶 Btwn ⟨𝑋, 𝐵⟩)
40		btwnconn1lem1 36068	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → ⟨𝐵, 𝑐⟩Cgr⟨𝑋, 𝐶⟩)
41		simp3lr 1246	. . . . 5 ⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩))) → ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩)
42	41	adantl 481	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩)
43	3, 5, 23, 24, 7, 6, 5, 37, 39, 40, 42	cgrextendand 35990	. . 3 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → ⟨𝐵, 𝑏⟩Cgr⟨𝑋, 𝐵⟩)
44	36, 43	jca 511	. 2 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → (𝐵 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝐵, 𝑏⟩Cgr⟨𝑋, 𝐵⟩))
45		segconeq 35991	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁))) → ((𝐴 ≠ 𝐵 ∧ (𝐵 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝐵, 𝑋⟩Cgr⟨𝑋, 𝐵⟩) ∧ (𝐵 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝐵, 𝑏⟩Cgr⟨𝑋, 𝐵⟩)) → 𝑋 = 𝑏))
46	3, 5, 7, 5, 4, 7, 24, 45	syl133anc 1395	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) → ((𝐴 ≠ 𝐵 ∧ (𝐵 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝐵, 𝑋⟩Cgr⟨𝑋, 𝐵⟩) ∧ (𝐵 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝐵, 𝑏⟩Cgr⟨𝑋, 𝐵⟩)) → 𝑋 = 𝑏))
47	46	adantr 480	. 2 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → ((𝐴 ≠ 𝐵 ∧ (𝐵 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝐵, 𝑋⟩Cgr⟨𝑋, 𝐵⟩) ∧ (𝐵 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝐵, 𝑏⟩Cgr⟨𝑋, 𝐵⟩)) → 𝑋 = 𝑏))
48	2, 22, 44, 47	mp3and 1466	1 ⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑋⟩ ∧ ⟨𝑑, 𝑋⟩Cgr⟨𝐷, 𝐵⟩)))) → 𝑋 = 𝑏)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ⟨cop 4591   class class class wbr 5102  ‘cfv 6499  ℕcn 12162  𝔼cee 28868   Btwn cbtwn 28869  Cgrccgr 28870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-oi 9439  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-z 12506  df-uz 12770  df-rp 12928  df-ico 13288  df-icc 13289  df-fz 13445  df-fzo 13592  df-seq 13943  df-exp 14003  df-hash 14272  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-clim 15430  df-sum 15629  df-ee 28871  df-btwn 28872  df-cgr 28873  df-ofs 35964

This theorem is referenced by:  btwnconn1lem14  36081