Theorem linethru 36441

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 36440	. . 3 ⊢ (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)(𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)))
2		simpll1 1222	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑛 ∈ ℕ)
3		simpll2 1223	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑎 ∈ (𝔼‘𝑛))
4		simpll3 1224	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑏 ∈ (𝔼‘𝑛))
5		simplr 776	. . . . . . . . . . . 12 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑎 ≠ 𝑏)
6		liness 36433	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏)) → (𝑎Line𝑏) ⊆ (𝔼‘𝑛))
7	2, 3, 4, 5, 6	syl13anc 1383	. . . . . . . . . . 11 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → (𝑎Line𝑏) ⊆ (𝔼‘𝑛))
8		simprll 786	. . . . . . . . . . 11 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑃 ∈ (𝑎Line𝑏))
9	7, 8	sseldd 3928	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑃 ∈ (𝔼‘𝑛))
10		simprlr 787	. . . . . . . . . . 11 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑄 ∈ (𝑎Line𝑏))
11	7, 10	sseldd 3928	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → 𝑄 ∈ (𝔼‘𝑛))
12		simplll 782	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎) ∧ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛))) → 𝑃 ∈ (𝑎Line𝑏))
13	12	adantl 484	. . . . . . . . . . . . . . 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 1222	. . . . . . . . . . . . . . . 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 1223	. . . . . . . . . . . . . . . 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 1224	. . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . 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 3002	. . . . . . . . . . . . . . . 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 36436	. . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . 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 36438	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑃)) → (𝑎Line𝑃) = (𝑃Line𝑎))
25	14, 15, 18, 20, 24	syl13anc 1383	. . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . 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 3010	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 = 𝑎 → (𝑃 ≠ 𝑄 ↔ 𝑃 ≠ 𝑎))
28	27	anbi2d 638	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 = 𝑎 → (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) ↔ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑎)))
29	28	anbi1d 639	. . . . . . . . . . . . . . 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 638	. . . . . . . . . . . . . 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 7389	. . . . . . . . . . . . . . 15 ⊢ (𝑄 = 𝑎 → (𝑃Line𝑄) = (𝑃Line𝑎))
32	31	eqeq2d 2763	. . . . . . . . . . . . . 14 ⊢ (𝑄 = 𝑎 → ((𝑎Line𝑏) = (𝑃Line𝑄) ↔ (𝑎Line𝑏) = (𝑃Line𝑎)))
33	30, 32	imbi12d 346	. . . . . . . . . . . . 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 260	. . . . . . . . . . . 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 1145	. . . . . . . . . . . . . . . . . 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 1209	. . . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . . . 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 1276	. . . . . . . . . . . . . . . . . 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 1277	. . . . . . . . . . . . . . . . . 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 1278	. . . . . . . . . . . . . . . . . 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 1208	. . . . . . . . . . . . . . . . . 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 1253	. . . . . . . . . . . . . . . . . 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 1147	. . . . . . . . . . . . . . . . . . 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 3002	. . . . . . . . . . . . . . . . . 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 36436	. . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . 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 36438	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑄 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑄)) → (𝑎Line𝑄) = (𝑄Line𝑎))
49	38, 39, 42, 44, 48	syl13anc 1383	. . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . 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 2854	. . . . . . . . . . . . . . . 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 1252	. . . . . . . . . . . . . . . . 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 1251	. . . . . . . . . . . . . . . . . 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 3002	. . . . . . . . . . . . . . . . 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 36436	. . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . 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 36438	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑛) ∧ 𝑃 ∈ (𝔼‘𝑛) ∧ 𝑄 ≠ 𝑃)) → (𝑄Line𝑃) = (𝑃Line𝑄))
60	38, 42, 53, 55, 59	syl13anc 1383	. . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . 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 1127	. . . . . . . . . . . . 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 416	. . . . . . . . . . . 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 3030	. . . . . . . . . . 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 459	. . . . . . . . . 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 707	. . . . . . . . 9 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) ∧ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)) → (𝑎Line𝑏) = (𝑃Line𝑄))
67	66	ex 415	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) → (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) → (𝑎Line𝑏) = (𝑃Line𝑄)))
68		eleq2 2841	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑎Line𝑏) → (𝑃 ∈ 𝐴 ↔ 𝑃 ∈ (𝑎Line𝑏)))
69		eleq2 2841	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑎Line𝑏) → (𝑄 ∈ 𝐴 ↔ 𝑄 ∈ (𝑎Line𝑏)))
70	68, 69	anbi12d 640	. . . . . . . . . 10 ⊢ (𝐴 = (𝑎Line𝑏) → ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ↔ (𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏))))
71	70	anbi1d 639	. . . . . . . . 9 ⊢ (𝐴 = (𝑎Line𝑏) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) ↔ ((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄)))
72		eqeq1 2756	. . . . . . . . 9 ⊢ (𝐴 = (𝑎Line𝑏) → (𝐴 = (𝑃Line𝑄) ↔ (𝑎Line𝑏) = (𝑃Line𝑄)))
73	71, 72	imbi12d 346	. . . . . . . 8 ⊢ (𝐴 = (𝑎Line𝑏) → ((((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄)) ↔ (((𝑃 ∈ (𝑎Line𝑏) ∧ 𝑄 ∈ (𝑎Line𝑏)) ∧ 𝑃 ≠ 𝑄) → (𝑎Line𝑏) = (𝑃Line𝑄))))
74	67, 73	syl5ibrcom 249	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) ∧ 𝑎 ≠ 𝑏) → (𝐴 = (𝑎Line𝑏) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
75	74	expimpd 456	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛)) → ((𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
76	75	3expa 1127	. . . . 5 ⊢ (((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛)) ∧ 𝑏 ∈ (𝔼‘𝑛)) → ((𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
77	76	rexlimdva 3153	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 𝑎 ∈ (𝔼‘𝑛)) → (∃𝑏 ∈ (𝔼‘𝑛)(𝑎 ≠ 𝑏 ∧ 𝐴 = (𝑎Line𝑏)) → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))))
78	77	rexlimivv 3194	. . 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 219	. 2 ⊢ (𝐴 ∈ LinesEE → (((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄)))
80	79	3impib 1125	1 ⊢ ((𝐴 ∈ LinesEE ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  ∃wrex 3076   ⊆ wss 3895  ‘cfv 6506  (class class class)co 7381  ℕcn 12196  𝔼cee 29023  Linecline2 36422  LinesEEclines2 36424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-inf2 9582  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-nel 3052  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-se 5590  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-isom 6515  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-er 8662  df-ec 8664  df-map 8794  df-en 8913  df-dom 8914  df-sdom 8915  df-fin 8916  df-sup 9374  df-oi 9444  df-card 9883  df-pnf 11204  df-mnf 11205  df-xr 11206  df-ltxr 11207  df-le 11208  df-sub 11402  df-neg 11403  df-div 11831  df-nn 12197  df-2 12266  df-3 12267  df-n0 12468  df-z 12555  df-uz 12826  df-rp 12980  df-ico 13341  df-icc 13342  df-fz 13499  df-fzo 13646  df-seq 14001  df-exp 14061  df-hash 14330  df-cj 15098  df-re 15099  df-im 15100  df-sqrt 15234  df-abs 15235  df-clim 15487  df-sum 15686  df-ee 29026  df-btwn 29027  df-cgr 29028  df-ofs 36271  df-colinear 36327  df-ifs 36328  df-cgr3 36329  df-fs 36330  df-line2 36425  df-lines2 36427

This theorem is referenced by:  hilbert1.2  36443  lineintmo  36445