Theorem linethru 34455

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 34454	. . 3 ⊢ (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)(𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)))
2		simpll1 1211	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑛 ∈ ℕ)
3		simpll2 1212	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑎 ∈ (𝔼‘𝑛))
4		simpll3 1213	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑏 ∈ (𝔼‘𝑛))
5		simplr 766	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑎 ≠ 𝑏)
6		liness 34447	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏)) → (𝑎Line𝑏) ⊆ (𝔼‘𝑛))
7	2, 3, 4, 5, 6	syl13anc 1371	. . . . . . . . . . 11 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → (𝑎Line𝑏) ⊆ (𝔼‘𝑛))
8		simprll 776	. . . . . . . . . . 11 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑃 ∈ (𝑎Line𝑏))
9	7, 8	sseldd 3922	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑃 ∈ (𝔼‘𝑛))
10		simprlr 777	. . . . . . . . . . 11 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑄 ∈ (𝑎Line𝑏))
11	7, 10	sseldd 3922	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑄 ∈ (𝔼‘𝑛))
12		simplll 772	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) → 𝑃 ∈ (𝑎Line𝑏))
13	12	adantl 482	. . . . . . . . . . . . . . 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 1211	. . . . . . . . . . . . . . . 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 1212	. . . . . . . . . . . . . . . 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 1213	. . . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . 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 2999	. . . . . . . . . . . . . . . 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 34450	. . . . . . . . . . . . . . . 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 1387	. . . . . . . . . . . . . . 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 34452	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑃)) → (𝑎Line𝑃) = (𝑃Line𝑎))
25	14, 15, 18, 20, 24	syl13anc 1371	. . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . 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 3007	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 = 𝑎 → (𝑃 ≠ 𝑄 ↔ 𝑃 ≠ 𝑎))
28	27	anbi2d 629	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 = 𝑎 → (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ↔ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎)))
29	28	anbi1d 630	. . . . . . . . . . . . . . 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 629	. . . . . . . . . . . . . 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 7283	. . . . . . . . . . . . . . 15 ⊢ (𝑄 = 𝑎 → (𝑃Line𝑄) = (𝑃Line𝑎))
32	31	eqeq2d 2749	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . 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 1135	. . . . . . . . . . . . . . . . . 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 1198	. . . . . . . . . . . . . . . . . 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 1265	. . . . . . . . . . . . . . . . . 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 1266	. . . . . . . . . . . . . . . . . 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 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) → 𝑏 ∈ (𝔼‘𝑛))
41		simp1r 1197	. . . . . . . . . . . . . . . . . 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 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) → 𝑄 ∈ (𝔼‘𝑛))
43		simp3 1137	. . . . . . . . . . . . . . . . . . 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 2999	. . . . . . . . . . . . . . . . . 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 34450	. . . . . . . . . . . . . . . . . 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 1387	. . . . . . . . . . . . . . . . 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 34452	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑄)) → (𝑎Line𝑄) = (𝑄Line𝑎))
49	38, 39, 42, 44, 48	syl13anc 1371	. . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . . 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 1241	. . . . . . . . . . . . . . . . 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 1240	. . . . . . . . . . . . . . . . . 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 2999	. . . . . . . . . . . . . . . . 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 34450	. . . . . . . . . . . . . . . . 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 1387	. . . . . . . . . . . . . . . 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 34452	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑛) ∧ 𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ≠ 𝑃)) → (𝑄Line𝑃) = (𝑃Line𝑄))
60	38, 42, 53, 55, 59	syl13anc 1371	. . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . 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 1117	. . . . . . . . . . . . 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 414	. . . . . . . . . . . 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 3028	. . . . . . . . . . 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 457	. . . . . . . . . 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 696	. . . . . . . . 9 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → (𝑎Line𝑏) = (𝑃Line𝑄))
67	66	ex 413	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) → (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) → (𝑎Line𝑏) = (𝑃Line𝑄)))
68		eleq2 2827	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑎Line𝑏) → (𝑃 ∈ 𝐴 ↔ 𝑃 ∈ (𝑎Line𝑏)))
69		eleq2 2827	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑎Line𝑏) → (𝑄 ∈ 𝐴 ↔ 𝑄 ∈ (𝑎Line𝑏)))
70	68, 69	anbi12d 631	. . . . . . . . . 10 ⊢ (𝐴 = (𝑎Line𝑏) → ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ↔ (𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏))))
71	70	anbi1d 630	. . . . . . . . 9 ⊢ (𝐴 = (𝑎Line𝑏) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ↔ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)))
72		eqeq1 2742	. . . . . . . . 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 454	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) → ((𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
76	75	3expa 1117	. . . . 5 ⊢ (((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛)) ∧ 𝑏 ∈ (𝔼‘𝑛)) → ((𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
77	76	rexlimdva 3213	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛)) → (∃𝑏 ∈ (𝔼‘𝑛)(𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
78	77	rexlimivv 3221	. . 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 1115	1 ⊢ ((𝐴 ∈ LinesEE ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3065   ⊆ wss 3887  ‘cfv 6433  (class class class)co 7275  ℕcn 11973  𝔼cee 27256  Linecline2 34436  LinesEEclines2 34438

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-ec 8500  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-sum 15398  df-ee 27259  df-btwn 27260  df-cgr 27261  df-ofs 34285  df-colinear 34341  df-ifs 34342  df-cgr3 34343  df-fs 34344  df-line2 34439  df-lines2 34441

This theorem is referenced by:  hilbert1.2  34457  lineintmo  34459