lineelsb2 - Mathbox for Scott Fenton

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Theorem lineelsb2 36893

Description: If 𝑆 lies on 𝑃𝑄, then 𝑃𝑄 = 𝑃𝑆. Theorem 6.16 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
lineelsb2	⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → (𝑆 ∈ (𝑃Line𝑄) → (𝑃Line𝑄) = (𝑃Line𝑆)))

Proof of Theorem lineelsb2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1210	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
2		simpl3l 1247	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑆 ∈ (𝔼‘𝑁))
3		simpl21 1270	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ∈ (𝔼‘𝑁))
4		simpl22 1271	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑄 ∈ (𝔼‘𝑁))
5		brcolinear 36804	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁))) → (𝑆 Colinear ⟨𝑃, 𝑄⟩ ↔ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑆⟩ ∨ 𝑄 Btwn ⟨𝑆, 𝑃⟩)))
6	1, 2, 3, 4, 5	syl13anc 1399	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑆 Colinear ⟨𝑃, 𝑄⟩ ↔ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑆⟩ ∨ 𝑄 Btwn ⟨𝑆, 𝑃⟩)))
7	6	biimpa 482	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Colinear ⟨𝑃, 𝑄⟩) → (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑆⟩ ∨ 𝑄 Btwn ⟨𝑆, 𝑃⟩))
8		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (𝔼‘𝑁))
9		brcolinear 36804	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁))) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ (𝑥 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑥⟩ ∨ 𝑄 Btwn ⟨𝑥, 𝑃⟩)))
10	1, 8, 3, 4, 9	syl13anc 1399	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ (𝑥 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑥⟩ ∨ 𝑄 Btwn ⟨𝑥, 𝑃⟩)))
11	10	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ (𝑥 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑥⟩ ∨ 𝑄 Btwn ⟨𝑥, 𝑃⟩)))
12		btwnconn3 36848	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑆 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁))) → ((𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩) → (𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩)))
13	1, 3, 2, 8, 4, 12	syl122anc 1406	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩) → (𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩)))
14	13	imp 412	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → (𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩))
15		btwncolinear3 36816	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑆 ∈ (𝔼‘𝑁))) → (𝑆 Btwn ⟨𝑃, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
16	1, 3, 8, 2, 15	syl13anc 1399	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑆 Btwn ⟨𝑃, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
17		btwncolinear5 36818	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑆 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → (𝑥 Btwn ⟨𝑃, 𝑆⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
18	1, 3, 2, 8, 17	syl13anc 1399	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑥 Btwn ⟨𝑃, 𝑆⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
19	16, 18	jaod 873	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩) → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
20	19	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → ((𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩) → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
21	14, 20	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → 𝑥 Colinear ⟨𝑃, 𝑆⟩)
22	21	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩) → (𝑥 Btwn ⟨𝑃, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
23		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑆 Btwn ⟨𝑃, 𝑄⟩)
24	1, 2, 3, 4, 23	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑆 Btwn ⟨𝑄, 𝑃⟩)
25		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑃 Btwn ⟨𝑄, 𝑥⟩)
26	1, 4, 2, 3, 8, 24, 25	btwnexch3and 36766	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑃 Btwn ⟨𝑆, 𝑥⟩)
27		btwncolinear4 36817	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → (𝑃 Btwn ⟨𝑆, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
28	1, 2, 8, 3, 27	syl13anc 1399	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑃 Btwn ⟨𝑆, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
29	28	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → (𝑃 Btwn ⟨𝑆, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
30	26, 29	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑥 Colinear ⟨𝑃, 𝑆⟩)
31	30	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩) → (𝑃 Btwn ⟨𝑄, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
32		simprl 783	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑆 Btwn ⟨𝑃, 𝑄⟩)
33		simprr 785	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑄 Btwn ⟨𝑥, 𝑃⟩)
34	1, 4, 8, 3, 33	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑄 Btwn ⟨𝑃, 𝑥⟩)
35	1, 3, 2, 4, 8, 32, 34	btwnexchand 36771	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑆 Btwn ⟨𝑃, 𝑥⟩)
36	16	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → (𝑆 Btwn ⟨𝑃, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
37	35, 36	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑥 Colinear ⟨𝑃, 𝑆⟩)
38	37	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩) → (𝑄 Btwn ⟨𝑥, 𝑃⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
39	22, 31, 38	3jaod 1456	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩) → ((𝑥 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑥⟩ ∨ 𝑄 Btwn ⟨𝑥, 𝑃⟩) → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
40	11, 39	sylbid 243	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
41		brcolinear 36804	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝑆 ∈ (𝔼‘𝑁))) → (𝑥 Colinear ⟨𝑃, 𝑆⟩ ↔ (𝑥 Btwn ⟨𝑃, 𝑆⟩ ∨ 𝑃 Btwn ⟨𝑆, 𝑥⟩ ∨ 𝑆 Btwn ⟨𝑥, 𝑃⟩)))
42	1, 8, 3, 2, 41	syl13anc 1399	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑥 Colinear ⟨𝑃, 𝑆⟩ ↔ (𝑥 Btwn ⟨𝑃, 𝑆⟩ ∨ 𝑃 Btwn ⟨𝑆, 𝑥⟩ ∨ 𝑆 Btwn ⟨𝑥, 𝑃⟩)))
43	42	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩) → (𝑥 Colinear ⟨𝑃, 𝑆⟩ ↔ (𝑥 Btwn ⟨𝑃, 𝑆⟩ ∨ 𝑃 Btwn ⟨𝑆, 𝑥⟩ ∨ 𝑆 Btwn ⟨𝑥, 𝑃⟩)))
44		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → 𝑥 Btwn ⟨𝑃, 𝑆⟩)
45		simprl 783	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → 𝑆 Btwn ⟨𝑃, 𝑄⟩)
46	1, 3, 8, 2, 4, 44, 45	btwnexchand 36771	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → 𝑥 Btwn ⟨𝑃, 𝑄⟩)
47		btwncolinear5 36818	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → (𝑥 Btwn ⟨𝑃, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
48	1, 3, 4, 8, 47	syl13anc 1399	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑥 Btwn ⟨𝑃, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
49	48	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → (𝑥 Btwn ⟨𝑃, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
50	46, 49	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → 𝑥 Colinear ⟨𝑃, 𝑄⟩)
51	50	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩) → (𝑥 Btwn ⟨𝑃, 𝑆⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
52		simpl3r 1248	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ≠ 𝑆)
53	52	necomd 3011	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑆 ≠ 𝑃)
54	53	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑆 ≠ 𝑃)
55		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑆 Btwn ⟨𝑃, 𝑄⟩)
56	1, 2, 3, 4, 55	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑆 Btwn ⟨𝑄, 𝑃⟩)
57		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑃 Btwn ⟨𝑆, 𝑥⟩)
58		btwnouttr2 36767	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝑆 ∈ (𝔼‘𝑁)) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → ((𝑆 ≠ 𝑃 ∧ 𝑆 Btwn ⟨𝑄, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩) → 𝑃 Btwn ⟨𝑄, 𝑥⟩))
59	1, 4, 2, 3, 8, 58	syl122anc 1406	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑆 ≠ 𝑃 ∧ 𝑆 Btwn ⟨𝑄, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩) → 𝑃 Btwn ⟨𝑄, 𝑥⟩))
60	59	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → ((𝑆 ≠ 𝑃 ∧ 𝑆 Btwn ⟨𝑄, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩) → 𝑃 Btwn ⟨𝑄, 𝑥⟩))
61	54, 56, 57, 60	mp3and 1493	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑃 Btwn ⟨𝑄, 𝑥⟩)
62		btwncolinear4 36817	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → (𝑃 Btwn ⟨𝑄, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
63	1, 4, 8, 3, 62	syl13anc 1399	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑃 Btwn ⟨𝑄, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
64	63	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → (𝑃 Btwn ⟨𝑄, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
65	61, 64	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑥 Colinear ⟨𝑃, 𝑄⟩)
66	65	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩) → (𝑃 Btwn ⟨𝑆, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
67	52	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑃 ≠ 𝑆)
68		simprl 783	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑆 Btwn ⟨𝑃, 𝑄⟩)
69		simprr 785	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑆 Btwn ⟨𝑥, 𝑃⟩)
70	1, 2, 8, 3, 69	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑆 Btwn ⟨𝑃, 𝑥⟩)
71		btwnconn1 36846	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑆 ∈ (𝔼‘𝑁)) ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → ((𝑃 ≠ 𝑆 ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑆 Btwn ⟨𝑃, 𝑥⟩) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))
72	1, 3, 2, 4, 8, 71	syl122anc 1406	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑃 ≠ 𝑆 ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑆 Btwn ⟨𝑃, 𝑥⟩) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))
73	72	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → ((𝑃 ≠ 𝑆 ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑆 Btwn ⟨𝑃, 𝑥⟩) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))
74	67, 68, 70, 73	mp3and 1493	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))
75		btwncolinear3 36816	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁))) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
76	1, 3, 8, 4, 75	syl13anc 1399	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
77	76, 48	jaod 873	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
78	77	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
79	74, 78	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑥 Colinear ⟨𝑃, 𝑄⟩)
80	79	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩) → (𝑆 Btwn ⟨𝑥, 𝑃⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
81	51, 66, 80	3jaod 1456	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩) → ((𝑥 Btwn ⟨𝑃, 𝑆⟩ ∨ 𝑃 Btwn ⟨𝑆, 𝑥⟩ ∨ 𝑆 Btwn ⟨𝑥, 𝑃⟩) → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
82	43, 81	sylbid 243	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩) → (𝑥 Colinear ⟨𝑃, 𝑆⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
83	40, 82	impbid 215	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Btwn ⟨𝑃, 𝑄⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ 𝑥 Colinear ⟨𝑃, 𝑆⟩))
84	10	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ (𝑥 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑥⟩ ∨ 𝑄 Btwn ⟨𝑥, 𝑃⟩)))
85		simprr 785	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → 𝑥 Btwn ⟨𝑃, 𝑄⟩)
86	1, 8, 3, 4, 85	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → 𝑥 Btwn ⟨𝑄, 𝑃⟩)
87		simprl 783	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → 𝑃 Btwn ⟨𝑄, 𝑆⟩)
88	1, 4, 8, 3, 2, 86, 87	btwnexch3and 36766	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → 𝑃 Btwn ⟨𝑥, 𝑆⟩)
89		btwncolinear2 36815	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → (𝑃 Btwn ⟨𝑥, 𝑆⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
90	1, 8, 2, 3, 89	syl13anc 1399	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑃 Btwn ⟨𝑥, 𝑆⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
91	90	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → (𝑃 Btwn ⟨𝑥, 𝑆⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
92	88, 91	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → 𝑥 Colinear ⟨𝑃, 𝑆⟩)
93	92	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩) → (𝑥 Btwn ⟨𝑃, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
94		simpl23 1272	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ≠ 𝑄)
95	94	necomd 3011	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑄 ≠ 𝑃)
96	95	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑄 ≠ 𝑃)
97		simprl 783	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑃 Btwn ⟨𝑄, 𝑆⟩)
98		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑃 Btwn ⟨𝑄, 𝑥⟩)
99		btwnconn2 36847	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁)) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → ((𝑄 ≠ 𝑃 ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩) → (𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩)))
100	1, 4, 3, 2, 8, 99	syl122anc 1406	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑄 ≠ 𝑃 ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩) → (𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩)))
101	100	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → ((𝑄 ≠ 𝑃 ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩) → (𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩)))
102	96, 97, 98, 101	mp3and 1493	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → (𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩))
103	19	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → ((𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩) → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
104	102, 103	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑥 Colinear ⟨𝑃, 𝑆⟩)
105	104	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩) → (𝑃 Btwn ⟨𝑄, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
106	94	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑃 ≠ 𝑄)
107		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑃 Btwn ⟨𝑄, 𝑆⟩)
108	1, 3, 4, 2, 107	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑃 Btwn ⟨𝑆, 𝑄⟩)
109		simprr 785	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑄 Btwn ⟨𝑥, 𝑃⟩)
110	1, 4, 8, 3, 109	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑄 Btwn ⟨𝑃, 𝑥⟩)
111		btwnouttr 36769	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁)) ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → ((𝑃 ≠ 𝑄 ∧ 𝑃 Btwn ⟨𝑆, 𝑄⟩ ∧ 𝑄 Btwn ⟨𝑃, 𝑥⟩) → 𝑃 Btwn ⟨𝑆, 𝑥⟩))
112	1, 2, 3, 4, 8, 111	syl122anc 1406	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑃 ≠ 𝑄 ∧ 𝑃 Btwn ⟨𝑆, 𝑄⟩ ∧ 𝑄 Btwn ⟨𝑃, 𝑥⟩) → 𝑃 Btwn ⟨𝑆, 𝑥⟩))
113	112	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → ((𝑃 ≠ 𝑄 ∧ 𝑃 Btwn ⟨𝑆, 𝑄⟩ ∧ 𝑄 Btwn ⟨𝑃, 𝑥⟩) → 𝑃 Btwn ⟨𝑆, 𝑥⟩))
114	106, 108, 110, 113	mp3and 1493	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑃 Btwn ⟨𝑆, 𝑥⟩)
115	28	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → (𝑃 Btwn ⟨𝑆, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
116	114, 115	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑥 Colinear ⟨𝑃, 𝑆⟩)
117	116	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩) → (𝑄 Btwn ⟨𝑥, 𝑃⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
118	93, 105, 117	3jaod 1456	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩) → ((𝑥 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑥⟩ ∨ 𝑄 Btwn ⟨𝑥, 𝑃⟩) → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
119	84, 118	sylbid 243	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
120	42	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩) → (𝑥 Colinear ⟨𝑃, 𝑆⟩ ↔ (𝑥 Btwn ⟨𝑃, 𝑆⟩ ∨ 𝑃 Btwn ⟨𝑆, 𝑥⟩ ∨ 𝑆 Btwn ⟨𝑥, 𝑃⟩)))
121		simprr 785	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → 𝑥 Btwn ⟨𝑃, 𝑆⟩)
122	1, 8, 3, 2, 121	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → 𝑥 Btwn ⟨𝑆, 𝑃⟩)
123		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → 𝑃 Btwn ⟨𝑄, 𝑆⟩)
124	1, 3, 4, 2, 123	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → 𝑃 Btwn ⟨𝑆, 𝑄⟩)
125	1, 2, 8, 3, 4, 122, 124	btwnexch3and 36766	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → 𝑃 Btwn ⟨𝑥, 𝑄⟩)
126		btwncolinear2 36815	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → (𝑃 Btwn ⟨𝑥, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
127	1, 8, 4, 3, 126	syl13anc 1399	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑃 Btwn ⟨𝑥, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
128	127	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → (𝑃 Btwn ⟨𝑥, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
129	125, 128	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → 𝑥 Colinear ⟨𝑃, 𝑄⟩)
130	129	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩) → (𝑥 Btwn ⟨𝑃, 𝑆⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
131	53	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑆 ≠ 𝑃)
132		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑃 Btwn ⟨𝑄, 𝑆⟩)
133	1, 3, 4, 2, 132	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑃 Btwn ⟨𝑆, 𝑄⟩)
134		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑃 Btwn ⟨𝑆, 𝑥⟩)
135		btwnconn2 36847	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁)) ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → ((𝑆 ≠ 𝑃 ∧ 𝑃 Btwn ⟨𝑆, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))
136	1, 2, 3, 4, 8, 135	syl122anc 1406	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑆 ≠ 𝑃 ∧ 𝑃 Btwn ⟨𝑆, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))
137	136	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → ((𝑆 ≠ 𝑃 ∧ 𝑃 Btwn ⟨𝑆, 𝑄⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))
138	131, 133, 134, 137	mp3and 1493	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))
139	77	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
140	138, 139	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑥 Colinear ⟨𝑃, 𝑄⟩)
141	140	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩) → (𝑃 Btwn ⟨𝑆, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
142	52	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑃 ≠ 𝑆)
143		simprl 783	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑃 Btwn ⟨𝑄, 𝑆⟩)
144		simprr 785	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑆 Btwn ⟨𝑥, 𝑃⟩)
145	1, 2, 8, 3, 144	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑆 Btwn ⟨𝑃, 𝑥⟩)
146		btwnouttr 36769	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁)) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → ((𝑃 ≠ 𝑆 ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑆 Btwn ⟨𝑃, 𝑥⟩) → 𝑃 Btwn ⟨𝑄, 𝑥⟩))
147	1, 4, 3, 2, 8, 146	syl122anc 1406	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑃 ≠ 𝑆 ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑆 Btwn ⟨𝑃, 𝑥⟩) → 𝑃 Btwn ⟨𝑄, 𝑥⟩))
148	147	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → ((𝑃 ≠ 𝑆 ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑆 Btwn ⟨𝑃, 𝑥⟩) → 𝑃 Btwn ⟨𝑄, 𝑥⟩))
149	142, 143, 145, 148	mp3and 1493	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑃 Btwn ⟨𝑄, 𝑥⟩)
150	63	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → (𝑃 Btwn ⟨𝑄, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
151	149, 150	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑆⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑥 Colinear ⟨𝑃, 𝑄⟩)
152	151	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩) → (𝑆 Btwn ⟨𝑥, 𝑃⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
153	130, 141, 152	3jaod 1456	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩) → ((𝑥 Btwn ⟨𝑃, 𝑆⟩ ∨ 𝑃 Btwn ⟨𝑆, 𝑥⟩ ∨ 𝑆 Btwn ⟨𝑥, 𝑃⟩) → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
154	120, 153	sylbid 243	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩) → (𝑥 Colinear ⟨𝑃, 𝑆⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
155	119, 154	impbid 215	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑆⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ 𝑥 Colinear ⟨𝑃, 𝑆⟩))
156	10	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ (𝑥 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑥⟩ ∨ 𝑄 Btwn ⟨𝑥, 𝑃⟩)))
157		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → 𝑥 Btwn ⟨𝑃, 𝑄⟩)
158		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → 𝑄 Btwn ⟨𝑆, 𝑃⟩)
159	1, 4, 2, 3, 158	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → 𝑄 Btwn ⟨𝑃, 𝑆⟩)
160	1, 3, 8, 4, 2, 157, 159	btwnexchand 36771	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → 𝑥 Btwn ⟨𝑃, 𝑆⟩)
161	18	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → (𝑥 Btwn ⟨𝑃, 𝑆⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
162	160, 161	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → 𝑥 Colinear ⟨𝑃, 𝑆⟩)
163	162	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩) → (𝑥 Btwn ⟨𝑃, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
164	95	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑄 ≠ 𝑃)
165		simprl 783	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑄 Btwn ⟨𝑆, 𝑃⟩)
166		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑃 Btwn ⟨𝑄, 𝑥⟩)
167		btwnouttr2 36767	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁)) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → ((𝑄 ≠ 𝑃 ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩) → 𝑃 Btwn ⟨𝑆, 𝑥⟩))
168	1, 2, 4, 3, 8, 167	syl122anc 1406	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑄 ≠ 𝑃 ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩) → 𝑃 Btwn ⟨𝑆, 𝑥⟩))
169	168	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → ((𝑄 ≠ 𝑃 ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩) → 𝑃 Btwn ⟨𝑆, 𝑥⟩))
170	164, 165, 166, 169	mp3and 1493	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑃 Btwn ⟨𝑆, 𝑥⟩)
171	28	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → (𝑃 Btwn ⟨𝑆, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
172	170, 171	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑥 Colinear ⟨𝑃, 𝑆⟩)
173	172	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩) → (𝑃 Btwn ⟨𝑄, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
174	94	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑃 ≠ 𝑄)
175		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑄 Btwn ⟨𝑆, 𝑃⟩)
176	1, 4, 2, 3, 175	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑄 Btwn ⟨𝑃, 𝑆⟩)
177		simprr 785	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑄 Btwn ⟨𝑥, 𝑃⟩)
178	1, 4, 8, 3, 177	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑄 Btwn ⟨𝑃, 𝑥⟩)
179		btwnconn1 36846	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁)) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → ((𝑃 ≠ 𝑄 ∧ 𝑄 Btwn ⟨𝑃, 𝑆⟩ ∧ 𝑄 Btwn ⟨𝑃, 𝑥⟩) → (𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩)))
180	1, 3, 4, 2, 8, 179	syl122anc 1406	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑃 ≠ 𝑄 ∧ 𝑄 Btwn ⟨𝑃, 𝑆⟩ ∧ 𝑄 Btwn ⟨𝑃, 𝑥⟩) → (𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩)))
181	180	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → ((𝑃 ≠ 𝑄 ∧ 𝑄 Btwn ⟨𝑃, 𝑆⟩ ∧ 𝑄 Btwn ⟨𝑃, 𝑥⟩) → (𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩)))
182	174, 176, 178, 181	mp3and 1493	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → (𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩))
183	19	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → ((𝑆 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑆⟩) → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
184	182, 183	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑄 Btwn ⟨𝑥, 𝑃⟩)) → 𝑥 Colinear ⟨𝑃, 𝑆⟩)
185	184	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩) → (𝑄 Btwn ⟨𝑥, 𝑃⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
186	163, 173, 185	3jaod 1456	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩) → ((𝑥 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑥⟩ ∨ 𝑄 Btwn ⟨𝑥, 𝑃⟩) → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
187	156, 186	sylbid 243	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑆⟩))
188	42	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩) → (𝑥 Colinear ⟨𝑃, 𝑆⟩ ↔ (𝑥 Btwn ⟨𝑃, 𝑆⟩ ∨ 𝑃 Btwn ⟨𝑆, 𝑥⟩ ∨ 𝑆 Btwn ⟨𝑥, 𝑃⟩)))
189		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → 𝑄 Btwn ⟨𝑆, 𝑃⟩)
190	1, 4, 2, 3, 189	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → 𝑄 Btwn ⟨𝑃, 𝑆⟩)
191		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → 𝑥 Btwn ⟨𝑃, 𝑆⟩)
192		btwnconn3 36848	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑆 ∈ (𝔼‘𝑁))) → ((𝑄 Btwn ⟨𝑃, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))
193	1, 3, 4, 8, 2, 192	syl122anc 1406	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑄 Btwn ⟨𝑃, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))
194	193	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → ((𝑄 Btwn ⟨𝑃, 𝑆⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))
195	190, 191, 194	mp2and 712	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))
196	77	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
197	195, 196	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑆⟩)) → 𝑥 Colinear ⟨𝑃, 𝑄⟩)
198	197	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩) → (𝑥 Btwn ⟨𝑃, 𝑆⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
199		simprl 783	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑄 Btwn ⟨𝑆, 𝑃⟩)
200		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑃 Btwn ⟨𝑆, 𝑥⟩)
201	1, 2, 4, 3, 8, 199, 200	btwnexch3and 36766	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑃 Btwn ⟨𝑄, 𝑥⟩)
202	63	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → (𝑃 Btwn ⟨𝑄, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
203	201, 202	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑆, 𝑥⟩)) → 𝑥 Colinear ⟨𝑃, 𝑄⟩)
204	203	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩) → (𝑃 Btwn ⟨𝑆, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
205		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑄 Btwn ⟨𝑆, 𝑃⟩)
206	1, 4, 2, 3, 205	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑄 Btwn ⟨𝑃, 𝑆⟩)
207		simprr 785	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑆 Btwn ⟨𝑥, 𝑃⟩)
208	1, 2, 8, 3, 207	btwncomand 36760	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑆 Btwn ⟨𝑃, 𝑥⟩)
209	1, 3, 4, 2, 8, 206, 208	btwnexchand 36771	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑄 Btwn ⟨𝑃, 𝑥⟩)
210	76	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
211	209, 210	mpd 16	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑄 Btwn ⟨𝑆, 𝑃⟩ ∧ 𝑆 Btwn ⟨𝑥, 𝑃⟩)) → 𝑥 Colinear ⟨𝑃, 𝑄⟩)
212	211	expr 462	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩) → (𝑆 Btwn ⟨𝑥, 𝑃⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
213	198, 204, 212	3jaod 1456	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩) → ((𝑥 Btwn ⟨𝑃, 𝑆⟩ ∨ 𝑃 Btwn ⟨𝑆, 𝑥⟩ ∨ 𝑆 Btwn ⟨𝑥, 𝑃⟩) → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
214	188, 213	sylbid 243	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩) → (𝑥 Colinear ⟨𝑃, 𝑆⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
215	187, 214	impbid 215	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑄 Btwn ⟨𝑆, 𝑃⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ 𝑥 Colinear ⟨𝑃, 𝑆⟩))
216	83, 155, 215	3jaodan 1458	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑆⟩ ∨ 𝑄 Btwn ⟨𝑆, 𝑃⟩)) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ 𝑥 Colinear ⟨𝑃, 𝑆⟩))
217	7, 216	syldan 603	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑆 Colinear ⟨𝑃, 𝑄⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ 𝑥 Colinear ⟨𝑃, 𝑆⟩))
218	217	adantrl 729	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑆 Colinear ⟨𝑃, 𝑄⟩)) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ 𝑥 Colinear ⟨𝑃, 𝑆⟩))
219	218	an32s 665	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑆 Colinear ⟨𝑃, 𝑄⟩)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ 𝑥 Colinear ⟨𝑃, 𝑆⟩))
220	219	rabbidva 3419	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑆 Colinear ⟨𝑃, 𝑄⟩)) → {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑄⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑆⟩})
221	220	ex 418	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → ((𝑆 ∈ (𝔼‘𝑁) ∧ 𝑆 Colinear ⟨𝑃, 𝑄⟩) → {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑄⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑆⟩}))
222		fvline2 36891	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → (𝑃Line𝑄) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑄⟩})
223	222	3adant3 1150	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → (𝑃Line𝑄) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑄⟩})
224	223	eleq2d 2847	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → (𝑆 ∈ (𝑃Line𝑄) ↔ 𝑆 ∈ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑄⟩}))
225		breq1 5106	. . . 4 ⊢ (𝑥 = 𝑆 → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ 𝑆 Colinear ⟨𝑃, 𝑄⟩))
226	225	elrab 3645	. . 3 ⊢ (𝑆 ∈ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑄⟩} ↔ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑆 Colinear ⟨𝑃, 𝑄⟩))
227	224, 226	bitrdi 290	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → (𝑆 ∈ (𝑃Line𝑄) ↔ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑆 Colinear ⟨𝑃, 𝑄⟩)))
228		simp1 1154	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → 𝑁 ∈ ℕ)
229		simp21 1225	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → 𝑃 ∈ (𝔼‘𝑁))
230		simp3l 1220	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → 𝑆 ∈ (𝔼‘𝑁))
231		simp3r 1221	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → 𝑃 ≠ 𝑆)
232		fvline2 36891	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → (𝑃Line𝑆) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑆⟩})
233	228, 229, 230, 231, 232	syl13anc 1399	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → (𝑃Line𝑆) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑆⟩})
234	223, 233	eqeq12d 2777	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → ((𝑃Line𝑄) = (𝑃Line𝑆) ↔ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑄⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑆⟩}))
235	221, 227, 234	3imtr4d 297	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → (𝑆 ∈ (𝑃Line𝑄) → (𝑃Line𝑄) = (𝑃Line𝑆)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 {crab 3413 ⟨cop 4590 class class class wbr 5103 ‘cfv 6537 (class class class)co 7418 ℕcn 12328 𝔼cee 29458 Btwn cbtwn 29459 Colinear ccolin 36782 Linecline2 36879

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-ec 8712 df-map 8842 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9427 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-n0 12600 df-z 12687 df-uz 12959 df-rp 13114 df-ico 13475 df-icc 13476 df-fz 13633 df-fzo 13782 df-seq 14138 df-exp 14198 df-hash 14468 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-clim 15648 df-sum 15847 df-ee 29461 df-btwn 29462 df-cgr 29463 df-ofs 36728 df-colinear 36784 df-ifs 36785 df-cgr3 36786 df-fs 36787 df-line2 36882

This theorem is used by: linethru 36898

W3C validator