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Theorem ellines 36134
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)(𝑝𝑞𝐴 = (𝑝Line𝑞)))
Distinct variable group:   𝐴,𝑛,𝑝,𝑞

Proof of Theorem ellines
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elex 3499 . 2 (𝐴 ∈ LinesEE → 𝐴 ∈ V)
2 ovex 7464 . . . . . . 7 (𝑝Line𝑞) ∈ V
3 eleq1 2827 . . . . . . 7 (𝐴 = (𝑝Line𝑞) → (𝐴 ∈ V ↔ (𝑝Line𝑞) ∈ V))
42, 3mpbiri 258 . . . . . 6 (𝐴 = (𝑝Line𝑞) → 𝐴 ∈ V)
54adantl 481 . . . . 5 ((𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
65rexlimivw 3149 . . . 4 (∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
76a1i 11 . . 3 ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V))
87rexlimivv 3199 . 2 (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
9 eleq1 2827 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ LinesEE ↔ 𝐴 ∈ LinesEE))
10 eqeq1 2739 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = (𝑝Line𝑞) ↔ 𝐴 = (𝑝Line𝑞)))
1110anbi2d 630 . . . . 5 (𝑥 = 𝐴 → ((𝑝𝑞𝑥 = (𝑝Line𝑞)) ↔ (𝑝𝑞𝐴 = (𝑝Line𝑞))))
1211rexbidv 3177 . . . 4 (𝑥 = 𝐴 → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)) ↔ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞))))
13122rexbidv 3220 . . 3 (𝑥 = 𝐴 → (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)) ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞))))
14 df-lines2 36121 . . . . . 6 LinesEE = ran Line
15 df-line2 36119 . . . . . . 7 Line = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )}
1615rneqi 5951 . . . . . 6 ran Line = ran {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )}
17 rnoprab 7537 . . . . . 6 ran {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )} = {𝑥 ∣ ∃𝑝𝑞𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )}
1814, 16, 173eqtri 2767 . . . . 5 LinesEE = {𝑥 ∣ ∃𝑝𝑞𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )}
1918eleq2i 2831 . . . 4 (𝑥 ∈ LinesEE ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑝𝑞𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )})
20 abid 2716 . . . . 5 (𝑥 ∈ {𝑥 ∣ ∃𝑝𝑞𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )} ↔ ∃𝑝𝑞𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear ))
21 df-rex 3069 . . . . . . 7 (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear ) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )))
22212exbii 1846 . . . . . 6 (∃𝑝𝑞𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear ) ↔ ∃𝑝𝑞𝑛(𝑛 ∈ ℕ ∧ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )))
23 exrot3 2163 . . . . . . 7 (∃𝑛𝑝𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))) ↔ ∃𝑝𝑞𝑛(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
24 r2ex 3194 . . . . . . . 8 (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)) ↔ ∃𝑛𝑝((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞))))
25 r19.42v 3189 . . . . . . . . . 10 (∃𝑞 ∈ (𝔼‘𝑛)((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞))))
26 df-rex 3069 . . . . . . . . . 10 (∃𝑞 ∈ (𝔼‘𝑛)((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ ∃𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
2725, 26bitr3i 277 . . . . . . . . 9 (((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ ∃𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
28272exbii 1846 . . . . . . . 8 (∃𝑛𝑝((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ ∃𝑛𝑝𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
2924, 28bitri 275 . . . . . . 7 (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)) ↔ ∃𝑛𝑝𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
30 anass 468 . . . . . . . . . 10 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ (𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
31 anass 468 . . . . . . . . . . 11 ((((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) ∧ 𝑥 = (𝑝Line𝑞)) ↔ ((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))))
32 simplrl 777 . . . . . . . . . . . . . 14 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → 𝑛 ∈ ℕ)
33 simplrr 778 . . . . . . . . . . . . . . 15 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → 𝑝 ∈ (𝔼‘𝑛))
34 simpll 767 . . . . . . . . . . . . . . 15 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → 𝑞 ∈ (𝔼‘𝑛))
35 simpr 484 . . . . . . . . . . . . . . 15 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → 𝑝𝑞)
3633, 34, 353jca 1127 . . . . . . . . . . . . . 14 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞))
3732, 36jca 511 . . . . . . . . . . . . 13 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → (𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)))
38 simpr2 1194 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → 𝑞 ∈ (𝔼‘𝑛))
39 simpl 482 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → 𝑛 ∈ ℕ)
40 simpr1 1193 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → 𝑝 ∈ (𝔼‘𝑛))
4138, 39, 40jca32 515 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → (𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))))
42 simpr3 1195 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → 𝑝𝑞)
4341, 42jca 511 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → ((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞))
4437, 43impbii 209 . . . . . . . . . . . 12 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) ↔ (𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)))
4544anbi1i 624 . . . . . . . . . . 11 ((((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) ∧ 𝑥 = (𝑝Line𝑞)) ↔ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) ∧ 𝑥 = (𝑝Line𝑞)))
4631, 45bitr3i 277 . . . . . . . . . 10 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) ∧ 𝑥 = (𝑝Line𝑞)))
4730, 46bitr3i 277 . . . . . . . . 9 ((𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))) ↔ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) ∧ 𝑥 = (𝑝Line𝑞)))
48 fvline 36126 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → (𝑝Line𝑞) = {𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩})
49 opex 5475 . . . . . . . . . . . . . 14 𝑝, 𝑞⟩ ∈ V
50 dfec2 8747 . . . . . . . . . . . . . 14 (⟨𝑝, 𝑞⟩ ∈ V → [⟨𝑝, 𝑞⟩] Colinear = {𝑥 ∣ ⟨𝑝, 𝑞 Colinear 𝑥})
5149, 50ax-mp 5 . . . . . . . . . . . . 13 [⟨𝑝, 𝑞⟩] Colinear = {𝑥 ∣ ⟨𝑝, 𝑞 Colinear 𝑥}
52 vex 3482 . . . . . . . . . . . . . . 15 𝑥 ∈ V
5349, 52brcnv 5896 . . . . . . . . . . . . . 14 (⟨𝑝, 𝑞 Colinear 𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩)
5453abbii 2807 . . . . . . . . . . . . 13 {𝑥 ∣ ⟨𝑝, 𝑞 Colinear 𝑥} = {𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩}
5551, 54eqtri 2763 . . . . . . . . . . . 12 [⟨𝑝, 𝑞⟩] Colinear = {𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩}
5648, 55eqtr4di 2793 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → (𝑝Line𝑞) = [⟨𝑝, 𝑞⟩] Colinear )
5756eqeq2d 2746 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → (𝑥 = (𝑝Line𝑞) ↔ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear ))
5857pm5.32i 574 . . . . . . . . 9 (((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) ∧ 𝑥 = (𝑝Line𝑞)) ↔ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear ))
59 anass 468 . . . . . . . . 9 (((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear ) ↔ (𝑛 ∈ ℕ ∧ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )))
6047, 58, 593bitrri 298 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )) ↔ (𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
61603exbii 1847 . . . . . . 7 (∃𝑝𝑞𝑛(𝑛 ∈ ℕ ∧ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )) ↔ ∃𝑝𝑞𝑛(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
6223, 29, 613bitr4ri 304 . . . . . 6 (∃𝑝𝑞𝑛(𝑛 ∈ ℕ ∧ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )) ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)))
6322, 62bitri 275 . . . . 5 (∃𝑝𝑞𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear ) ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)))
6420, 63bitri 275 . . . 4 (𝑥 ∈ {𝑥 ∣ ∃𝑝𝑞𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )} ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)))
6519, 64bitri 275 . . 3 (𝑥 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)))
669, 13, 65vtoclbg 3557 . 2 (𝐴 ∈ V → (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞))))
671, 8, 66pm5.21nii 378 1 (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  wrex 3068  Vcvv 3478  cop 4637   class class class wbr 5148  ccnv 5688  ran crn 5690  cfv 6563  (class class class)co 7431  {coprab 7432  [cec 8742  cn 12264  𝔼cee 28918   Colinear ccolin 36019  Linecline2 36116  LinesEEclines2 36118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-ec 8746  df-nn 12265  df-colinear 36021  df-line2 36119  df-lines2 36121
This theorem is referenced by:  linethru  36135  hilbert1.1  36136
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