Theorem linethru 35063

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 35062	. . 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 35055	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏)) → (𝑎Line𝑏) ⊆ (𝔼‘𝑛))
7	2, 3, 4, 5, 6	syl13anc 1373	. . . . . . . . . . 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 3982	. . . . . . . . . 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 3982	. . . . . . . . . 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 483	. . . . . . . . . . . . . . 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 2997	. . . . . . . . . . . . . . . 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 35058	. . . . . . . . . . . . . . . 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 1389	. . . . . . . . . . . . . . 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 35060	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑃)) → (𝑎Line𝑃) = (𝑃Line𝑎))
25	14, 15, 18, 20, 24	syl13anc 1373	. . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . 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 3005	. . . . . . . . . . . . . . . . 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 7412	. . . . . . . . . . . . . . 15 ⊢ (𝑄 = 𝑎 → (𝑃Line𝑄) = (𝑃Line𝑎))
32	31	eqeq2d 2744	. . . . . . . . . . . . . 14 ⊢ (𝑄 = 𝑎 → ((𝑎Line𝑏) = (𝑃Line𝑄) ↔ (𝑎Line𝑏) = (𝑃Line𝑎)))
33	30, 32	imbi12d 345	. . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . . . . . . . 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 2997	. . . . . . . . . . . . . . . . . 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 35058	. . . . . . . . . . . . . . . . . 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 1389	. . . . . . . . . . . . . . . . 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 35060	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑄)) → (𝑎Line𝑄) = (𝑄Line𝑎))
49	38, 39, 42, 44, 48	syl13anc 1373	. . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . . . . 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 2997	. . . . . . . . . . . . . . . . 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 35058	. . . . . . . . . . . . . . . . 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 1389	. . . . . . . . . . . . . . . 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 35060	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑛) ∧ 𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ≠ 𝑃)) → (𝑄Line𝑃) = (𝑃Line𝑄))
60	38, 42, 53, 55, 59	syl13anc 1373	. . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 3026	. . . . . . . . . . 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 458	. . . . . . . . . 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 698	. . . . . . . . 9 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → (𝑎Line𝑏) = (𝑃Line𝑄))
67	66	ex 414	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) → (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) → (𝑎Line𝑏) = (𝑃Line𝑄)))
68		eleq2 2823	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑎Line𝑏) → (𝑃 ∈ 𝐴 ↔ 𝑃 ∈ (𝑎Line𝑏)))
69		eleq2 2823	. . . . . . . . . . 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 2737	. . . . . . . . 9 ⊢ (𝐴 = (𝑎Line𝑏) → (𝐴 = (𝑃Line𝑄) ↔ (𝑎Line𝑏) = (𝑃Line𝑄)))
73	71, 72	imbi12d 345	. . . . . . . 8 ⊢ (𝐴 = (𝑎Line𝑏) → ((((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄)) ↔ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) → (𝑎Line𝑏) = (𝑃Line𝑄))))
74	67, 73	syl5ibrcom 246	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) → (𝐴 = (𝑎Line𝑏) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
75	74	expimpd 455	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) → ((𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
76	75	3expa 1119	. . . . 5 ⊢ (((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛)) ∧ 𝑏 ∈ (𝔼‘𝑛)) → ((𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
77	76	rexlimdva 3156	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛)) → (∃𝑏 ∈ (𝔼‘𝑛)(𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
78	77	rexlimivv 3200	. . 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 216	. 2 ⊢ (𝐴 ∈ LinesEE → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄)))
80	79	3impib 1117	1 ⊢ ((𝐴 ∈ LinesEE ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∃wrex 3071   ⊆ wss 3947  ‘cfv 6540  (class class class)co 7404  ℕcn 12208  𝔼cee 28126  Linecline2 35044  LinesEEclines2 35046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-ec 8701  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-ico 13326  df-icc 13327  df-fz 13481  df-fzo 13624  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-sum 15629  df-ee 28129  df-btwn 28130  df-cgr 28131  df-ofs 34893  df-colinear 34949  df-ifs 34950  df-cgr3 34951  df-fs 34952  df-line2 35047  df-lines2 35049

This theorem is referenced by:  hilbert1.2  35065  lineintmo  35067