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Theorem ellines 31928
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 3201 . 2 (𝐴 ∈ LinesEE → 𝐴 ∈ V)
2 ovex 6638 . . . . . . 7 (𝑝Line𝑞) ∈ V
3 eleq1 2686 . . . . . . 7 (𝐴 = (𝑝Line𝑞) → (𝐴 ∈ V ↔ (𝑝Line𝑞) ∈ V))
42, 3mpbiri 248 . . . . . 6 (𝐴 = (𝑝Line𝑞) → 𝐴 ∈ V)
54adantl 482 . . . . 5 ((𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
65rexlimivw 3023 . . . 4 (∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
76a1i 11 . . 3 ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V))
87rexlimivv 3030 . 2 (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
9 eleq1 2686 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ LinesEE ↔ 𝐴 ∈ LinesEE))
10 eqeq1 2625 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = (𝑝Line𝑞) ↔ 𝐴 = (𝑝Line𝑞)))
1110anbi2d 739 . . . . 5 (𝑥 = 𝐴 → ((𝑝𝑞𝑥 = (𝑝Line𝑞)) ↔ (𝑝𝑞𝐴 = (𝑝Line𝑞))))
1211rexbidv 3046 . . . 4 (𝑥 = 𝐴 → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)) ↔ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞))))
13122rexbidv 3051 . . 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 31915 . . . . . 6 LinesEE = ran Line
15 df-line2 31913 . . . . . . 7 Line = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )}
1615rneqi 5317 . . . . . 6 ran Line = ran {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )}
17 rnoprab 6703 . . . . . 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 2647 . . . . 5 LinesEE = {𝑥 ∣ ∃𝑝𝑞𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )}
1918eleq2i 2690 . . . 4 (𝑥 ∈ LinesEE ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑝𝑞𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )})
20 abid 2609 . . . . 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 2913 . . . . . . 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 1772 . . . . . 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 2042 . . . . . . 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 3055 . . . . . . . 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 3085 . . . . . . . . . 10 (∃𝑞 ∈ (𝔼‘𝑛)((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞))))
26 df-rex 2913 . . . . . . . . . 10 (∃𝑞 ∈ (𝔼‘𝑛)((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ ∃𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
2725, 26bitr3i 266 . . . . . . . . 9 (((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ ∃𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
28272exbii 1772 . . . . . . . 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 264 . . . . . . 7 (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)) ↔ ∃𝑛𝑝𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
30 anass 680 . . . . . . . . . 10 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ (𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
31 anass 680 . . . . . . . . . . 11 ((((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) ∧ 𝑥 = (𝑝Line𝑞)) ↔ ((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))))
32 simplrl 799 . . . . . . . . . . . . . 14 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → 𝑛 ∈ ℕ)
33 simplrr 800 . . . . . . . . . . . . . . 15 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → 𝑝 ∈ (𝔼‘𝑛))
34 simpll 789 . . . . . . . . . . . . . . 15 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → 𝑞 ∈ (𝔼‘𝑛))
35 simpr 477 . . . . . . . . . . . . . . 15 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → 𝑝𝑞)
3633, 34, 353jca 1240 . . . . . . . . . . . . . 14 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞))
3732, 36jca 554 . . . . . . . . . . . . 13 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → (𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)))
38 simpr2 1066 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → 𝑞 ∈ (𝔼‘𝑛))
39 simpl 473 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → 𝑛 ∈ ℕ)
40 simpr1 1065 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → 𝑝 ∈ (𝔼‘𝑛))
4138, 39, 40jca32 557 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → (𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))))
42 simpr3 1067 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → 𝑝𝑞)
4341, 42jca 554 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → ((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞))
4437, 43impbii 199 . . . . . . . . . . . 12 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) ↔ (𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)))
4544anbi1i 730 . . . . . . . . . . 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 266 . . . . . . . . . 10 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) ∧ 𝑥 = (𝑝Line𝑞)))
4730, 46bitr3i 266 . . . . . . . . 9 ((𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))) ↔ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) ∧ 𝑥 = (𝑝Line𝑞)))
48 fvline 31920 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → (𝑝Line𝑞) = {𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩})
49 opex 4898 . . . . . . . . . . . . . 14 𝑝, 𝑞⟩ ∈ V
50 dfec2 7697 . . . . . . . . . . . . . 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 3192 . . . . . . . . . . . . . . 15 𝑥 ∈ V
5349, 52brcnv 5270 . . . . . . . . . . . . . 14 (⟨𝑝, 𝑞 Colinear 𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩)
5453abbii 2736 . . . . . . . . . . . . 13 {𝑥 ∣ ⟨𝑝, 𝑞 Colinear 𝑥} = {𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩}
5551, 54eqtri 2643 . . . . . . . . . . . 12 [⟨𝑝, 𝑞⟩] Colinear = {𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩}
5648, 55syl6eqr 2673 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → (𝑝Line𝑞) = [⟨𝑝, 𝑞⟩] Colinear )
5756eqeq2d 2631 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → (𝑥 = (𝑝Line𝑞) ↔ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear ))
5857pm5.32i 668 . . . . . . . . 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 680 . . . . . . . . 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 287 . . . . . . . 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 1773 . . . . . . 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 293 . . . . . 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 264 . . . . 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 264 . . . 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 264 . . 3 (𝑥 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)))
669, 13, 65vtoclbg 3256 . 2 (𝐴 ∈ V → (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞))))
671, 8, 66pm5.21nii 368 1 (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wrex 2908  Vcvv 3189  cop 4159   class class class wbr 4618  ccnv 5078  ran crn 5080  cfv 5852  (class class class)co 6610  {coprab 6611  [cec 7692  cn 10971  𝔼cee 25681   Colinear ccolin 31813  Linecline2 31910  LinesEEclines2 31912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-i2m1 9955  ax-1ne0 9956  ax-rrecex 9959  ax-cnre 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-ec 7696  df-nn 10972  df-colinear 31815  df-line2 31913  df-lines2 31915
This theorem is referenced by:  linethru  31929  hilbert1.1  31930
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