ellines - Mathbox for Scott Fenton

	Mathbox for Scott Fenton		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ellines		Structured version Visualization version GIF version

Theorem ellines 35646

Description: Membership in the set of all lines. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
ellines	⊢ (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)))

Distinct variable group: 𝐴,𝑛,𝑝,𝑞

Proof of Theorem ellines Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3485	. 2 ⊢ (𝐴 ∈ LinesEE → 𝐴 ∈ V)
2		ovex 7435	. . . . . . 7 ⊢ (𝑝Line𝑞) ∈ V
3		eleq1 2813	. . . . . . 7 ⊢ (𝐴 = (𝑝Line𝑞) → (𝐴 ∈ V ↔ (𝑝Line𝑞) ∈ V))
4	2, 3	mpbiri 258	. . . . . 6 ⊢ (𝐴 = (𝑝Line𝑞) → 𝐴 ∈ V)
5	4	adantl 481	. . . . 5 ⊢ ((𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
6	5	rexlimivw 3143	. . . 4 ⊢ (∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
7	6	a1i 11	. . 3 ⊢ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V))
8	7	rexlimivv 3191	. 2 ⊢ (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
9		eleq1 2813	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ LinesEE ↔ 𝐴 ∈ LinesEE))
10		eqeq1 2728	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = (𝑝Line𝑞) ↔ 𝐴 = (𝑝Line𝑞)))
11	10	anbi2d 628	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)) ↔ (𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))))
12	11	rexbidv 3170	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)) ↔ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))))
13	12	2rexbidv 3211	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)) ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))))
14		df-lines2 35633	. . . . . 6 ⊢ LinesEE = ran Line
15		df-line2 35631	. . . . . . 7 ⊢ Line = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )}
16	15	rneqi 5927	. . . . . 6 ⊢ ran Line = ran {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )}
17		rnoprab 7505	. . . . . 6 ⊢ ran {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )} = {𝑥 ∣ ∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )}
18	14, 16, 17	3eqtri 2756	. . . . 5 ⊢ LinesEE = {𝑥 ∣ ∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )}
19	18	eleq2i 2817	. . . 4 ⊢ (𝑥 ∈ LinesEE ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )})
20		abid 2705	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )} ↔ ∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear ))
21		df-rex 3063	. . . . . . 7 ⊢ (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear ) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )))
22	21	2exbii 1843	. . . . . 6 ⊢ (∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear ) ↔ ∃𝑝∃𝑞∃𝑛(𝑛 ∈ ℕ ∧ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )))
23		exrot3 2157	. . . . . . 7 ⊢ (∃𝑛∃𝑝∃𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))) ↔ ∃𝑝∃𝑞∃𝑛(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))))
24		r2ex 3187	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)) ↔ ∃𝑛∃𝑝((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞))))
25		r19.42v 3182	. . . . . . . . . 10 ⊢ (∃𝑞 ∈ (𝔼‘𝑛)((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞))) ↔ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞))))
26		df-rex 3063	. . . . . . . . . 10 ⊢ (∃𝑞 ∈ (𝔼‘𝑛)((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞))) ↔ ∃𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))))
27	25, 26	bitr3i 277	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞))) ↔ ∃𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))))
28	27	2exbii 1843	. . . . . . . 8 ⊢ (∃𝑛∃𝑝((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞))) ↔ ∃𝑛∃𝑝∃𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))))
29	24, 28	bitri 275	. . . . . . 7 ⊢ (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)) ↔ ∃𝑛∃𝑝∃𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))))
30		anass 468	. . . . . . . . . 10 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞))) ↔ (𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))))
31		anass 468	. . . . . . . . . . 11 ⊢ ((((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = (𝑝Line𝑞)) ↔ ((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞))))
32		simplrl 774	. . . . . . . . . . . . . 14 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑛 ∈ ℕ)
33		simplrr 775	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑝 ∈ (𝔼‘𝑛))
34		simpll 764	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑞 ∈ (𝔼‘𝑛))
35		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑝 ≠ 𝑞)
36	33, 34, 35	3jca 1125	. . . . . . . . . . . . . 14 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞))
37	32, 36	jca 511	. . . . . . . . . . . . 13 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → (𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)))
38		simpr2 1192	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑞 ∈ (𝔼‘𝑛))
39		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑛 ∈ ℕ)
40		simpr1 1191	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑝 ∈ (𝔼‘𝑛))
41	38, 39, 40	jca32 515	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))))
42		simpr3 1193	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑝 ≠ 𝑞)
43	41, 42	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → ((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞))
44	37, 43	impbii 208	. . . . . . . . . . . 12 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) ↔ (𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)))
45	44	anbi1i 623	. . . . . . . . . . 11 ⊢ ((((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = (𝑝Line𝑞)) ↔ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) ∧ 𝑥 = (𝑝Line𝑞)))
46	31, 45	bitr3i 277	. . . . . . . . . 10 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞))) ↔ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) ∧ 𝑥 = (𝑝Line𝑞)))
47	30, 46	bitr3i 277	. . . . . . . . 9 ⊢ ((𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))) ↔ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) ∧ 𝑥 = (𝑝Line𝑞)))
48		fvline 35638	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑝Line𝑞) = {𝑥 ∣ 𝑥 Colinear ⟨𝑝, 𝑞⟩})
49		opex 5455	. . . . . . . . . . . . . 14 ⊢ ⟨𝑝, 𝑞⟩ ∈ V
50		dfec2 8703	. . . . . . . . . . . . . 14 ⊢ (⟨𝑝, 𝑞⟩ ∈ V → [⟨𝑝, 𝑞⟩]^◡ Colinear = {𝑥 ∣ ⟨𝑝, 𝑞⟩^◡ Colinear 𝑥})
51	49, 50	ax-mp 5	. . . . . . . . . . . . 13 ⊢ [⟨𝑝, 𝑞⟩]^◡ Colinear = {𝑥 ∣ ⟨𝑝, 𝑞⟩^◡ Colinear 𝑥}
52		vex 3470	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
53	49, 52	brcnv 5873	. . . . . . . . . . . . . 14 ⊢ (⟨𝑝, 𝑞⟩^◡ Colinear 𝑥 ↔ 𝑥 Colinear ⟨𝑝, 𝑞⟩)
54	53	abbii 2794	. . . . . . . . . . . . 13 ⊢ {𝑥 ∣ ⟨𝑝, 𝑞⟩^◡ Colinear 𝑥} = {𝑥 ∣ 𝑥 Colinear ⟨𝑝, 𝑞⟩}
55	51, 54	eqtri 2752	. . . . . . . . . . . 12 ⊢ [⟨𝑝, 𝑞⟩]^◡ Colinear = {𝑥 ∣ 𝑥 Colinear ⟨𝑝, 𝑞⟩}
56	48, 55	eqtr4di 2782	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑝Line𝑞) = [⟨𝑝, 𝑞⟩]^◡ Colinear )
57	56	eqeq2d 2735	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑥 = (𝑝Line𝑞) ↔ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear ))
58	57	pm5.32i 574	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) ∧ 𝑥 = (𝑝Line𝑞)) ↔ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear ))
59		anass 468	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear ) ↔ (𝑛 ∈ ℕ ∧ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )))
60	47, 58, 59	3bitrri 298	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )) ↔ (𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))))
61	60	3exbii 1844	. . . . . . 7 ⊢ (∃𝑝∃𝑞∃𝑛(𝑛 ∈ ℕ ∧ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )) ↔ ∃𝑝∃𝑞∃𝑛(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))))
62	23, 29, 61	3bitr4ri 304	. . . . . 6 ⊢ (∃𝑝∃𝑞∃𝑛(𝑛 ∈ ℕ ∧ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )) ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))
63	22, 62	bitri 275	. . . . 5 ⊢ (∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear ) ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))
64	20, 63	bitri 275	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )} ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))
65	19, 64	bitri 275	. . 3 ⊢ (𝑥 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))
66	9, 13, 65	vtoclbg 3537	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))))
67	1, 8, 66	pm5.21nii 378	1 ⊢ (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2701 ≠ wne 2932 ∃wrex 3062 Vcvv 3466 ⟨cop 4627 class class class wbr 5139 ^◡ccnv 5666 ran crn 5668 ‘cfv 6534 (class class class)co 7402 {coprab 7403 [cec 8698 ℕcn 12211 𝔼cee 28639 Colinear ccolin 35531 Linecline2 35628 LinesEEclines2 35630

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-1cn 11165 ax-addcl 11167

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-ec 8702 df-nn 12212 df-colinear 35533 df-line2 35631 df-lines2 35633

This theorem is referenced by: linethru 35647 hilbert1.1 35648

W3C validator