Theorem linethru 36347

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 36346	. . 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 36339	. . . . . . . . . . . 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 3934	. . . . . . . . . 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 3934	. . . . . . . . . 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 2987	. . . . . . . . . . . . . . . 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 36342	. . . . . . . . . . . . . . . 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 36344	. . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 2995	. . . . . . . . . . . . . . . . 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 7366	. . . . . . . . . . . . . . 15 ⊢ (𝑄 = 𝑎 → (𝑃Line𝑄) = (𝑃Line𝑎))
32	31	eqeq2d 2747	. . . . . . . . . . . . . 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 2987	. . . . . . . . . . . . . . . . . 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 36342	. . . . . . . . . . . . . . . . . 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 36344	. . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . . . 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 2987	. . . . . . . . . . . . . . . . 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 36342	. . . . . . . . . . . . . . . . 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 36344	. . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . 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 3015	. . . . . . . . . . 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 2825	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑎Line𝑏) → (𝑃 ∈ 𝐴 ↔ 𝑃 ∈ (𝑎Line𝑏)))
69		eleq2 2825	. . . . . . . . . . 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 2740	. . . . . . . . 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 3137	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛)) → (∃𝑏 ∈ (𝔼‘𝑛)(𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
78	77	rexlimivv 3178	. . 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 1541   ∈ wcel 2113   ≠ wne 2932  ∃wrex 3060   ⊆ wss 3901  ‘cfv 6492  (class class class)co 7358  ℕcn 12145  𝔼cee 28960  Linecline2 36328  LinesEEclines2 36330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-ec 8637  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-n0 12402  df-z 12489  df-uz 12752  df-rp 12906  df-ico 13267  df-icc 13268  df-fz 13424  df-fzo 13571  df-seq 13925  df-exp 13985  df-hash 14254  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-clim 15411  df-sum 15610  df-ee 28963  df-btwn 28964  df-cgr 28965  df-ofs 36177  df-colinear 36233  df-ifs 36234  df-cgr3 36235  df-fs 36236  df-line2 36331  df-lines2 36333

This theorem is referenced by:  hilbert1.2  36349  lineintmo  36351