ellines - Mathbox for Scott Fenton

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Theorem ellines 34029

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 3428	. 2 ⊢ (𝐴 ∈ LinesEE → 𝐴 ∈ V)
2		ovex 7188	. . . . . . 7 ⊢ (𝑝Line𝑞) ∈ V
3		eleq1 2839	. . . . . . 7 ⊢ (𝐴 = (𝑝Line𝑞) → (𝐴 ∈ V ↔ (𝑝Line𝑞) ∈ V))
4	2, 3	mpbiri 261	. . . . . 6 ⊢ (𝐴 = (𝑝Line𝑞) → 𝐴 ∈ V)
5	4	adantl 485	. . . . 5 ⊢ ((𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
6	5	rexlimivw 3206	. . . 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 3216	. 2 ⊢ (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
9		eleq1 2839	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ LinesEE ↔ 𝐴 ∈ LinesEE))
10		eqeq1 2762	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = (𝑝Line𝑞) ↔ 𝐴 = (𝑝Line𝑞)))
11	10	anbi2d 631	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)) ↔ (𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))))
12	11	rexbidv 3221	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)) ↔ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))))
13	12	2rexbidv 3224	. . 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 34016	. . . . . 6 ⊢ LinesEE = ran Line
15		df-line2 34014	. . . . . . 7 ⊢ Line = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )}
16	15	rneqi 5782	. . . . . 6 ⊢ ran Line = ran {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )}
17		rnoprab 7256	. . . . . 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 2785	. . . . 5 ⊢ LinesEE = {𝑥 ∣ ∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )}
19	18	eleq2i 2843	. . . 4 ⊢ (𝑥 ∈ LinesEE ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )})
20		abid 2739	. . . . 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 3076	. . . . . . 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 1850	. . . . . 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 2169	. . . . . . 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 3227	. . . . . . . 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 3268	. . . . . . . . . 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 3076	. . . . . . . . . 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 280	. . . . . . . . 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 1850	. . . . . . . 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 278	. . . . . . 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 472	. . . . . . . . . 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 472	. . . . . . . . . . 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 776	. . . . . . . . . . . . . 14 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑛 ∈ ℕ)
33		simplrr 777	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑝 ∈ (𝔼‘𝑛))
34		simpll 766	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑞 ∈ (𝔼‘𝑛))
35		simpr 488	. . . . . . . . . . . . . . 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 515	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 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 519	. . . . . . . . . . . . . 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 515	. . . . . . . . . . . . 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 212	. . . . . . . . . . . 12 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) ↔ (𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)))
45	44	anbi1i 626	. . . . . . . . . . 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 280	. . . . . . . . . 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 280	. . . . . . . . 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 34021	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑝Line𝑞) = {𝑥 ∣ 𝑥 Colinear ⟨𝑝, 𝑞⟩})
49		opex 5327	. . . . . . . . . . . . . 14 ⊢ ⟨𝑝, 𝑞⟩ ∈ V
50		dfec2 8307	. . . . . . . . . . . . . 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 3413	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
53	49, 52	brcnv 5727	. . . . . . . . . . . . . 14 ⊢ (⟨𝑝, 𝑞⟩^◡ Colinear 𝑥 ↔ 𝑥 Colinear ⟨𝑝, 𝑞⟩)
54	53	abbii 2823	. . . . . . . . . . . . 13 ⊢ {𝑥 ∣ ⟨𝑝, 𝑞⟩^◡ Colinear 𝑥} = {𝑥 ∣ 𝑥 Colinear ⟨𝑝, 𝑞⟩}
55	51, 54	eqtri 2781	. . . . . . . . . . . 12 ⊢ [⟨𝑝, 𝑞⟩]^◡ Colinear = {𝑥 ∣ 𝑥 Colinear ⟨𝑝, 𝑞⟩}
56	48, 55	eqtr4di 2811	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑝Line𝑞) = [⟨𝑝, 𝑞⟩]^◡ Colinear )
57	56	eqeq2d 2769	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑥 = (𝑝Line𝑞) ↔ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear ))
58	57	pm5.32i 578	. . . . . . . . 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 472	. . . . . . . . 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 301	. . . . . . . 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 1851	. . . . . . 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 307	. . . . . 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 278	. . . . 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 278	. . . 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 278	. . 3 ⊢ (𝑥 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))
66	9, 13, 65	vtoclbg 3489	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))))
67	1, 8, 66	pm5.21nii 383	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 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2735 ≠ wne 2951 ∃wrex 3071 Vcvv 3409 ⟨cop 4531 class class class wbr 5035 ^◡ccnv 5526 ran crn 5528 ‘cfv 6339 (class class class)co 7155 {coprab 7156 [cec 8302 ℕcn 11679 𝔼cee 26786 Colinear ccolin 33914 Linecline2 34011 LinesEEclines2 34013

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-cnex 10636 ax-1cn 10638 ax-addcl 10640

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7158 df-oprab 7159 df-om 7585 df-wrecs 7962 df-recs 8023 df-rdg 8061 df-ec 8306 df-nn 11680 df-colinear 33916 df-line2 34014 df-lines2 34016

This theorem is referenced by: linethru 34030 hilbert1.1 34031

W3C validator