ellines - Mathbox for Scott Fenton

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Theorem ellines 36740

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 3474	. 2 ⊢ (𝐴 ∈ LinesEE → 𝐴 ∈ V)
2		ovex 7450	. . . . . . 7 ⊢ (𝑝Line𝑞) ∈ V
3		eleq1 2850	. . . . . . 7 ⊢ (𝐴 = (𝑝Line𝑞) → (𝐴 ∈ V ↔ (𝑝Line𝑞) ∈ V))
4	2, 3	mpbiri 261	. . . . . 6 ⊢ (𝐴 = (𝑝Line𝑞) → 𝐴 ∈ V)
5	4	adantl 487	. . . . 5 ⊢ ((𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
6	5	rexlimivw 3161	. . . 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 3206	. 2 ⊢ (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
9		eleq1 2850	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ LinesEE ↔ 𝐴 ∈ LinesEE))
10		eqeq1 2766	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = (𝑝Line𝑞) ↔ 𝐴 = (𝑝Line𝑞)))
11	10	anbi2d 642	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)) ↔ (𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))))
12	11	rexbidv 3188	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)) ↔ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))))
13	12	2rexbidv 3229	. . 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 36727	. . . . . 6 ⊢ LinesEE = ran Line
15		df-line2 36725	. . . . . . 7 ⊢ Line = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )}
16	15	rneqi 5925	. . . . . 6 ⊢ ran Line = ran {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )}
17		rnoprab 7522	. . . . . 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 2789	. . . . 5 ⊢ LinesEE = {𝑥 ∣ ∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )}
19	18	eleq2i 2854	. . . 4 ⊢ (𝑥 ∈ LinesEE ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )})
20		abid 2744	. . . . 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 3089	. . . . . . 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 1882	. . . . . 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 2202	. . . . . . 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 3201	. . . . . . . 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 3196	. . . . . . . . . 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 3089	. . . . . . . . . 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 1882	. . . . . . . 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 474	. . . . . . . . . 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 474	. . . . . . . . . . 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 789	. . . . . . . . . . . . . 14 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑛 ∈ ℕ)
33		simplrr 790	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑝 ∈ (𝔼‘𝑛))
34		simpll 779	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑞 ∈ (𝔼‘𝑛))
35		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑝 ≠ 𝑞)
36	33, 34, 35	3jca 1146	. . . . . . . . . . . . . 14 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞))
37	32, 36	jca 521	. . . . . . . . . . . . 13 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → (𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)))
38		simpr2 1214	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑞 ∈ (𝔼‘𝑛))
39		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑛 ∈ ℕ)
40		simpr1 1213	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑝 ∈ (𝔼‘𝑛))
41	38, 39, 40	jca32 525	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))))
42		simpr3 1215	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑝 ≠ 𝑞)
43	41, 42	jca 521	. . . . . . . . . . . . 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 636	. . . . . . . . . . 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 36732	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑝Line𝑞) = {𝑥 ∣ 𝑥 Colinear ⟨𝑝, 𝑞⟩})
49		opex 5443	. . . . . . . . . . . . . 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 3457	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
53	49, 52	brcnv 5866	. . . . . . . . . . . . . 14 ⊢ (⟨𝑝, 𝑞⟩^◡ Colinear 𝑥 ↔ 𝑥 Colinear ⟨𝑝, 𝑞⟩)
54	53	abbii 2829	. . . . . . . . . . . . 13 ⊢ {𝑥 ∣ ⟨𝑝, 𝑞⟩^◡ Colinear 𝑥} = {𝑥 ∣ 𝑥 Colinear ⟨𝑝, 𝑞⟩}
55	51, 54	eqtri 2785	. . . . . . . . . . . 12 ⊢ [⟨𝑝, 𝑞⟩]^◡ Colinear = {𝑥 ∣ 𝑥 Colinear ⟨𝑝, 𝑞⟩}
56	48, 55	eqtr4di 2815	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑝Line𝑞) = [⟨𝑝, 𝑞⟩]^◡ Colinear )
57	56	eqeq2d 2773	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑥 = (𝑝Line𝑞) ↔ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear ))
58	57	pm5.32i 585	. . . . . . . . 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 474	. . . . . . . . 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 1883	. . . . . . 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 3522	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))))
67	1, 8, 66	pm5.21nii 381	1 ⊢ (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2145 {cab 2740 ≠ wne 2957 ∃wrex 3088 Vcvv 3453 ⟨cop 4593 class class class wbr 5107 ^◡ccnv 5658 ran crn 5660 ‘cfv 6537 (class class class)co 7417 {coprab 7418 [cec 8698 ℕcn 12261 𝔼cee 29352 Colinear ccolin 36625 Linecline2 36722 LinesEEclines2 36724

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-1cn 11186 ax-addcl 11188

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7420 df-oprab 7421 df-om 7867 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-ec 8702 df-nn 12262 df-colinear 36627 df-line2 36725 df-lines2 36727

This theorem is used by: linethru 36741 hilbert1.1 36742

W3C validator