ellines - Mathbox for Scott Fenton

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Theorem ellines 34737

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 3463	. 2 ⊢ (𝐴 ∈ LinesEE → 𝐴 ∈ V)
2		ovex 7390	. . . . . . 7 ⊢ (𝑝Line𝑞) ∈ V
3		eleq1 2825	. . . . . . 7 ⊢ (𝐴 = (𝑝Line𝑞) → (𝐴 ∈ V ↔ (𝑝Line𝑞) ∈ V))
4	2, 3	mpbiri 257	. . . . . 6 ⊢ (𝐴 = (𝑝Line𝑞) → 𝐴 ∈ V)
5	4	adantl 482	. . . . 5 ⊢ ((𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
6	5	rexlimivw 3148	. . . 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 3196	. 2 ⊢ (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
9		eleq1 2825	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ LinesEE ↔ 𝐴 ∈ LinesEE))
10		eqeq1 2740	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = (𝑝Line𝑞) ↔ 𝐴 = (𝑝Line𝑞)))
11	10	anbi2d 629	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)) ↔ (𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))))
12	11	rexbidv 3175	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)) ↔ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))))
13	12	2rexbidv 3213	. . 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 34724	. . . . . 6 ⊢ LinesEE = ran Line
15		df-line2 34722	. . . . . . 7 ⊢ Line = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )}
16	15	rneqi 5892	. . . . . 6 ⊢ ran Line = ran {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )}
17		rnoprab 7460	. . . . . 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 2768	. . . . 5 ⊢ LinesEE = {𝑥 ∣ ∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )}
19	18	eleq2i 2829	. . . 4 ⊢ (𝑥 ∈ LinesEE ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear )})
20		abid 2717	. . . . 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 3074	. . . . . . 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 1851	. . . . . 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 2165	. . . . . . 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 3192	. . . . . . . 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 3187	. . . . . . . . . 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 3074	. . . . . . . . . 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 276	. . . . . . . . 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 1851	. . . . . . . 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 274	. . . . . . 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 469	. . . . . . . . . 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 469	. . . . . . . . . . 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 775	. . . . . . . . . . . . . 14 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑛 ∈ ℕ)
33		simplrr 776	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑝 ∈ (𝔼‘𝑛))
34		simpll 765	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑞 ∈ (𝔼‘𝑛))
35		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑝 ≠ 𝑞)
36	33, 34, 35	3jca 1128	. . . . . . . . . . . . . 14 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞))
37	32, 36	jca 512	. . . . . . . . . . . . 13 ⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → (𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)))
38		simpr2 1195	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑞 ∈ (𝔼‘𝑛))
39		simpl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑛 ∈ ℕ)
40		simpr1 1194	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑝 ∈ (𝔼‘𝑛))
41	38, 39, 40	jca32 516	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))))
42		simpr3 1196	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑝 ≠ 𝑞)
43	41, 42	jca 512	. . . . . . . . . . . . 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 624	. . . . . . . . . . 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 276	. . . . . . . . . 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 276	. . . . . . . . 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 34729	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑝Line𝑞) = {𝑥 ∣ 𝑥 Colinear ⟨𝑝, 𝑞⟩})
49		opex 5421	. . . . . . . . . . . . . 14 ⊢ ⟨𝑝, 𝑞⟩ ∈ V
50		dfec2 8651	. . . . . . . . . . . . . 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 3449	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
53	49, 52	brcnv 5838	. . . . . . . . . . . . . 14 ⊢ (⟨𝑝, 𝑞⟩^◡ Colinear 𝑥 ↔ 𝑥 Colinear ⟨𝑝, 𝑞⟩)
54	53	abbii 2806	. . . . . . . . . . . . 13 ⊢ {𝑥 ∣ ⟨𝑝, 𝑞⟩^◡ Colinear 𝑥} = {𝑥 ∣ 𝑥 Colinear ⟨𝑝, 𝑞⟩}
55	51, 54	eqtri 2764	. . . . . . . . . . . 12 ⊢ [⟨𝑝, 𝑞⟩]^◡ Colinear = {𝑥 ∣ 𝑥 Colinear ⟨𝑝, 𝑞⟩}
56	48, 55	eqtr4di 2794	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑝Line𝑞) = [⟨𝑝, 𝑞⟩]^◡ Colinear )
57	56	eqeq2d 2747	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑥 = (𝑝Line𝑞) ↔ 𝑥 = [⟨𝑝, 𝑞⟩]^◡ Colinear ))
58	57	pm5.32i 575	. . . . . . . . 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 469	. . . . . . . . 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 297	. . . . . . . 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 1852	. . . . . . 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 303	. . . . . 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 274	. . . . 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 274	. . . 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 274	. . 3 ⊢ (𝑥 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)))
66	9, 13, 65	vtoclbg 3528	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))))
67	1, 8, 66	pm5.21nii 379	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 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2713 ≠ wne 2943 ∃wrex 3073 Vcvv 3445 ⟨cop 4592 class class class wbr 5105 ^◡ccnv 5632 ran crn 5634 ‘cfv 6496 (class class class)co 7357 {coprab 7358 [cec 8646 ℕcn 12153 𝔼cee 27837 Colinear ccolin 34622 Linecline2 34719 LinesEEclines2 34721

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-1cn 11109 ax-addcl 11111

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-ec 8650 df-nn 12154 df-colinear 34624 df-line2 34722 df-lines2 34724

This theorem is referenced by: linethru 34738 hilbert1.1 34739

W3C validator