Theorem linethru 36138

Description: If 𝐴 is a line containing two distinct points 𝑃 and 𝑄, then 𝐴 is the line through 𝑃 and 𝑄. Theorem 6.18 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
linethru	⊢ ((𝐴 ∈ LinesEE ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))

Proof of Theorem linethru

Dummy variables 𝑎 𝑏 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ellines 36137	. . 3 ⊢ (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)(𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)))
2		simpll1 1213	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑛 ∈ ℕ)
3		simpll2 1214	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑎 ∈ (𝔼‘𝑛))
4		simpll3 1215	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑏 ∈ (𝔼‘𝑛))
5		simplr 768	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑎 ≠ 𝑏)
6		liness 36130	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏)) → (𝑎Line𝑏) ⊆ (𝔼‘𝑛))
7	2, 3, 4, 5, 6	syl13anc 1374	. . . . . . . . . . 11 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → (𝑎Line𝑏) ⊆ (𝔼‘𝑛))
8		simprll 778	. . . . . . . . . . 11 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑃 ∈ (𝑎Line𝑏))
9	7, 8	sseldd 3955	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑃 ∈ (𝔼‘𝑛))
10		simprlr 779	. . . . . . . . . . 11 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑄 ∈ (𝑎Line𝑏))
11	7, 10	sseldd 3955	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑄 ∈ (𝔼‘𝑛))
12		simplll 774	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) → 𝑃 ∈ (𝑎Line𝑏))
13	12	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → 𝑃 ∈ (𝑎Line𝑏))
14		simpll1 1213	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → 𝑛 ∈ ℕ)
15		simpll2 1214	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → 𝑎 ∈ (𝔼‘𝑛))
16		simpll3 1215	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → 𝑏 ∈ (𝔼‘𝑛))
17		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → 𝑎 ≠ 𝑏)
18		simprrl 780	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → 𝑃 ∈ (𝔼‘𝑛))
19		simprlr 779	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → 𝑃 ≠ 𝑎)
20	19	necomd 2982	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → 𝑎 ≠ 𝑃)
21		lineelsb2 36133	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑃)) → (𝑃 ∈ (𝑎Line𝑏) → (𝑎Line𝑏) = (𝑎Line𝑃)))
22	14, 15, 16, 17, 18, 20, 21	syl132anc 1390	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → (𝑃 ∈ (𝑎Line𝑏) → (𝑎Line𝑏) = (𝑎Line𝑃)))
23	13, 22	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → (𝑎Line𝑏) = (𝑎Line𝑃))
24		linecom 36135	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑃)) → (𝑎Line𝑃) = (𝑃Line𝑎))
25	14, 15, 18, 20, 24	syl13anc 1374	. . . . . . . . . . . . . 14 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → (𝑎Line𝑃) = (𝑃Line𝑎))
26	23, 25	eqtrd 2765	. . . . . . . . . . . . 13 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → (𝑎Line𝑏) = (𝑃Line𝑎))
27		neeq2 2990	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 = 𝑎 → (𝑃 ≠ 𝑄 ↔ 𝑃 ≠ 𝑎))
28	27	anbi2d 630	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 = 𝑎 → (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ↔ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎)))
29	28	anbi1d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑄 = 𝑎 → ((((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ↔ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))))
30	29	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑄 = 𝑎 → ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) ↔ (((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))))))
31		oveq2 7402	. . . . . . . . . . . . . . 15 ⊢ (𝑄 = 𝑎 → (𝑃Line𝑄) = (𝑃Line𝑎))
32	31	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (𝑄 = 𝑎 → ((𝑎Line𝑏) = (𝑃Line𝑄) ↔ (𝑎Line𝑏) = (𝑃Line𝑎)))
33	30, 32	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑄 = 𝑎 → (((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → (𝑎Line𝑏) = (𝑃Line𝑄)) ↔ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → (𝑎Line𝑏) = (𝑃Line𝑎))))
34	26, 33	mpbiri 258	. . . . . . . . . . . 12 ⊢ (𝑄 = 𝑎 → ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → (𝑎Line𝑏) = (𝑃Line𝑄)))
35		simp1 1136	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → ((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏))
36		simp2l 1200	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄))
37	35, 36, 10	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → 𝑄 ∈ (𝑎Line𝑏))
38		simp1l1 1267	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → 𝑛 ∈ ℕ)
39		simp1l2 1268	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → 𝑎 ∈ (𝔼‘𝑛))
40		simp1l3 1269	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → 𝑏 ∈ (𝔼‘𝑛))
41		simp1r 1199	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → 𝑎 ≠ 𝑏)
42		simp2rr 1244	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → 𝑄 ∈ (𝔼‘𝑛))
43		simp3 1138	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → 𝑄 ≠ 𝑎)
44	43	necomd 2982	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → 𝑎 ≠ 𝑄)
45		lineelsb2 36133	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℕ ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ (𝑄 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑄)) → (𝑄 ∈ (𝑎Line𝑏) → (𝑎Line𝑏) = (𝑎Line𝑄)))
46	38, 39, 40, 41, 42, 44, 45	syl132anc 1390	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → (𝑄 ∈ (𝑎Line𝑏) → (𝑎Line𝑏) = (𝑎Line𝑄)))
47	37, 46	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → (𝑎Line𝑏) = (𝑎Line𝑄))
48		linecom 36135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑄)) → (𝑎Line𝑄) = (𝑄Line𝑎))
49	38, 39, 42, 44, 48	syl13anc 1374	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → (𝑎Line𝑄) = (𝑄Line𝑎))
50	47, 49	eqtrd 2765	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → (𝑎Line𝑏) = (𝑄Line𝑎))
51	36	simplld 767	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → 𝑃 ∈ (𝑎Line𝑏))
52	51, 50	eleqtrd 2831	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → 𝑃 ∈ (𝑄Line𝑎))
53		simp2rl 1243	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → 𝑃 ∈ (𝔼‘𝑛))
54		simp2lr 1242	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → 𝑃 ≠ 𝑄)
55	54	necomd 2982	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → 𝑄 ≠ 𝑃)
56		lineelsb2 36133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑄 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ≠ 𝑃)) → (𝑃 ∈ (𝑄Line𝑎) → (𝑄Line𝑎) = (𝑄Line𝑃)))
57	38, 42, 39, 43, 53, 55, 56	syl132anc 1390	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → (𝑃 ∈ (𝑄Line𝑎) → (𝑄Line𝑎) = (𝑄Line𝑃)))
58	52, 57	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → (𝑄Line𝑎) = (𝑄Line𝑃))
59		linecom 36135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑛) ∧ 𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ≠ 𝑃)) → (𝑄Line𝑃) = (𝑃Line𝑄))
60	38, 42, 53, 55, 59	syl13anc 1374	. . . . . . . . . . . . . . 15 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → (𝑄Line𝑃) = (𝑃Line𝑄))
61	50, 58, 60	3eqtrd 2769	. . . . . . . . . . . . . 14 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) ∧ 𝑄 ≠ 𝑎) → (𝑎Line𝑏) = (𝑃Line𝑄))
62	61	3expa 1118	. . . . . . . . . . . . 13 ⊢ (((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) ∧ 𝑄 ≠ 𝑎) → (𝑎Line𝑏) = (𝑃Line𝑄))
63	62	expcom 413	. . . . . . . . . . . 12 ⊢ (𝑄 ≠ 𝑎 → ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → (𝑎Line𝑏) = (𝑃Line𝑄)))
64	34, 63	pm2.61ine 3010	. . . . . . . . . . 11 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)))) → (𝑎Line𝑏) = (𝑃Line𝑄))
65	64	expr 456	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛)) → (𝑎Line𝑏) = (𝑃Line𝑄)))
66	9, 11, 65	mp2and 699	. . . . . . . . 9 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → (𝑎Line𝑏) = (𝑃Line𝑄))
67	66	ex 412	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) → (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) → (𝑎Line𝑏) = (𝑃Line𝑄)))
68		eleq2 2818	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑎Line𝑏) → (𝑃 ∈ 𝐴 ↔ 𝑃 ∈ (𝑎Line𝑏)))
69		eleq2 2818	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑎Line𝑏) → (𝑄 ∈ 𝐴 ↔ 𝑄 ∈ (𝑎Line𝑏)))
70	68, 69	anbi12d 632	. . . . . . . . . 10 ⊢ (𝐴 = (𝑎Line𝑏) → ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ↔ (𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏))))
71	70	anbi1d 631	. . . . . . . . 9 ⊢ (𝐴 = (𝑎Line𝑏) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ↔ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)))
72		eqeq1 2734	. . . . . . . . 9 ⊢ (𝐴 = (𝑎Line𝑏) → (𝐴 = (𝑃Line𝑄) ↔ (𝑎Line𝑏) = (𝑃Line𝑄)))
73	71, 72	imbi12d 344	. . . . . . . 8 ⊢ (𝐴 = (𝑎Line𝑏) → ((((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄)) ↔ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) → (𝑎Line𝑏) = (𝑃Line𝑄))))
74	67, 73	syl5ibrcom 247	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) → (𝐴 = (𝑎Line𝑏) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
75	74	expimpd 453	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) → ((𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
76	75	3expa 1118	. . . . 5 ⊢ (((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛)) ∧ 𝑏 ∈ (𝔼‘𝑛)) → ((𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
77	76	rexlimdva 3136	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛)) → (∃𝑏 ∈ (𝔼‘𝑛)(𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
78	77	rexlimivv 3181	. . 3 ⊢ (∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)(𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄)))
79	1, 78	sylbi 217	. 2 ⊢ (𝐴 ∈ LinesEE → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄)))
80	79	3impib 1116	1 ⊢ ((𝐴 ∈ LinesEE ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2927  ∃wrex 3055   ⊆ wss 3922  ‘cfv 6519  (class class class)co 7394  ℕcn 12197  𝔼cee 28822  Linecline2 36119  LinesEEclines2 36121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718  ax-inf2 9612  ax-cnex 11142  ax-resscn 11143  ax-1cn 11144  ax-icn 11145  ax-addcl 11146  ax-addrcl 11147  ax-mulcl 11148  ax-mulrcl 11149  ax-mulcom 11150  ax-addass 11151  ax-mulass 11152  ax-distr 11153  ax-i2m1 11154  ax-1ne0 11155  ax-1rid 11156  ax-rnegex 11157  ax-rrecex 11158  ax-cnre 11159  ax-pre-lttri 11160  ax-pre-lttrn 11161  ax-pre-ltadd 11162  ax-pre-mulgt0 11163  ax-pre-sup 11164

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-nel 3032  df-ral 3047  df-rex 3056  df-rmo 3357  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-pss 3942  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-int 4919  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5541  df-eprel 5546  df-po 5554  df-so 5555  df-fr 5599  df-se 5600  df-we 5601  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-isom 6528  df-riota 7351  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7851  df-1st 7977  df-2nd 7978  df-frecs 8269  df-wrecs 8300  df-recs 8349  df-rdg 8387  df-1o 8443  df-er 8682  df-ec 8684  df-map 8805  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-sup 9411  df-oi 9481  df-card 9910  df-pnf 11228  df-mnf 11229  df-xr 11230  df-ltxr 11231  df-le 11232  df-sub 11425  df-neg 11426  df-div 11852  df-nn 12198  df-2 12260  df-3 12261  df-n0 12459  df-z 12546  df-uz 12810  df-rp 12966  df-ico 13325  df-icc 13326  df-fz 13482  df-fzo 13629  df-seq 13977  df-exp 14037  df-hash 14306  df-cj 15075  df-re 15076  df-im 15077  df-sqrt 15211  df-abs 15212  df-clim 15461  df-sum 15660  df-ee 28825  df-btwn 28826  df-cgr 28827  df-ofs 35968  df-colinear 36024  df-ifs 36025  df-cgr3 36026  df-fs 36027  df-line2 36122  df-lines2 36124

This theorem is referenced by:  hilbert1.2  36140  lineintmo  36142