lineelsb2 - Mathbox for Scott Fenton

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Theorem lineelsb2 36346

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 1193	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
2		simpl3l 1230	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑆 ∈ (𝔼‘𝑁))
3		simpl21 1253	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ∈ (𝔼‘𝑁))
4		simpl22 1254	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑄 ∈ (𝔼‘𝑁))
5		brcolinear 36257	. . . . . . . . 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 1375	. . . . . . . 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 476	. . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (𝔼‘𝑁))
9		brcolinear 36257	. . . . . . . . . . . 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 1375	. . . . . . . . . . 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 480	. . . . . . . . . 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 36301	. . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . 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 406	. . . . . . . . . . . . 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 36269	. . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . 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 36271	. . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . 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 860	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 771	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . 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 36219	. . . . . . . . . . . . 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 36270	. . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 36224	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 1432	. . . . . . . . . 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 240	. . . . . . . . 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 36257	. . . . . . . . . . . 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 1375	. . . . . . . . . . 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 480	. . . . . . . . . 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 773	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 36224	. . . . . . . . . . . . 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 36271	. . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 1231	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ≠ 𝑆)
53	52	necomd 2988	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑆 ≠ 𝑃)
54	53	adantr 480	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . 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 36220	. . . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 1467	. . . . . . . . . . . . 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 36270	. . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 36299	. . . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 1467	. . . . . . . . . . . . 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 36269	. . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . 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 860	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 1432	. . . . . . . . . 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 240	. . . . . . . . 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 212	. . . . . . . 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 480	. . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 36219	. . . . . . . . . . . . 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 36268	. . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 1255	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ≠ 𝑄)
95	94	necomd 2988	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑄 ≠ 𝑃)
96	95	adantr 480	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . 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 36300	. . . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 1467	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 36222	. . . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 1467	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 1432	. . . . . . . . . 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 240	. . . . . . . . 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 480	. . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 36219	. . . . . . . . . . . . 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 36268	. . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . 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 36300	. . . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 1467	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 36222	. . . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 1467	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 1432	. . . . . . . . . 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 240	. . . . . . . . 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 212	. . . . . . . 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 480	. . . . . . . . . 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 773	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 36224	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . 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 36220	. . . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 1467	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 36299	. . . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 1467	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 1432	. . . . . . . . . 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 240	. . . . . . . . 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 480	. . . . . . . . . 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 771	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . 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 36301	. . . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 700	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . 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 36219	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 771	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 36213	. . . . . . . . . . . . . 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 36224	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 1432	. . . . . . . . . 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 240	. . . . . . . . 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 212	. . . . . . . 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 1434	. . . . . . 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 592	. . . . . 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 717	. . . . 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 653	. . . 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 3396	. . 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 412	. 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 36344	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → (𝑃Line𝑄) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑄⟩})
223	222	3adant3 1133	. . . 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 2823	. . 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 5089	. . . 4 ⊢ (𝑥 = 𝑆 → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ 𝑆 Colinear ⟨𝑃, 𝑄⟩))
226	225	elrab 3635	. . 3 ⊢ (𝑆 ∈ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑄⟩} ↔ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑆 Colinear ⟨𝑃, 𝑄⟩))
227	224, 226	bitrdi 287	. 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 1137	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → 𝑁 ∈ ℕ)
229		simp21 1208	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → 𝑃 ∈ (𝔼‘𝑁))
230		simp3l 1203	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → 𝑆 ∈ (𝔼‘𝑁))
231		simp3r 1204	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → 𝑃 ≠ 𝑆)
232		fvline2 36344	. . . 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 1375	. . 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 2753	. 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 294	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

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∨ w3o 1086 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 {crab 3390 ⟨cop 4574 class class class wbr 5086 ‘cfv 6492 (class class class)co 7360 ℕcn 12165 𝔼cee 28970 Btwn cbtwn 28971 Colinear ccolin 36235 Linecline2 36332

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-ec 8638 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9348 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-sum 15640 df-ee 28973 df-btwn 28974 df-cgr 28975 df-ofs 36181 df-colinear 36237 df-ifs 36238 df-cgr3 36239 df-fs 36240 df-line2 36335

This theorem is referenced by: linethru 36351

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