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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)(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 3428 . 2 (𝐴 ∈ LinesEE → 𝐴 ∈ V)
2 ovex 7188 . . . . . . 7 (𝑝Line𝑞) ∈ V
3 eleq1 2839 . . . . . . 7 (𝐴 = (𝑝Line𝑞) → (𝐴 ∈ V ↔ (𝑝Line𝑞) ∈ V))
42, 3mpbiri 261 . . . . . 6 (𝐴 = (𝑝Line𝑞) → 𝐴 ∈ V)
54adantl 485 . . . . 5 ((𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
65rexlimivw 3206 . . . 4 (∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
76a1i 11 . . 3 ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V))
87rexlimivv 3216 . 2 (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
9 eleq1 2839 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ LinesEE ↔ 𝐴 ∈ LinesEE))
10 eqeq1 2762 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = (𝑝Line𝑞) ↔ 𝐴 = (𝑝Line𝑞)))
1110anbi2d 631 . . . . 5 (𝑥 = 𝐴 → ((𝑝𝑞𝑥 = (𝑝Line𝑞)) ↔ (𝑝𝑞𝐴 = (𝑝Line𝑞))))
1211rexbidv 3221 . . . 4 (𝑥 = 𝐴 → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)) ↔ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞))))
13122rexbidv 3224 . . 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 34016 . . . . . 6 LinesEE = ran Line
15 df-line2 34014 . . . . . . 7 Line = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )}
1615rneqi 5782 . . . . . 6 ran Line = ran {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )}
17 rnoprab 7256 . . . . . 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 2785 . . . . 5 LinesEE = {𝑥 ∣ ∃𝑝𝑞𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )}
1918eleq2i 2843 . . . 4 (𝑥 ∈ LinesEE ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑝𝑞𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )})
20 abid 2739 . . . . 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 3076 . . . . . . 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 1850 . . . . . 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 2169 . . . . . . 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 3227 . . . . . . . 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 3268 . . . . . . . . . 10 (∃𝑞 ∈ (𝔼‘𝑛)((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞))))
26 df-rex 3076 . . . . . . . . . 10 (∃𝑞 ∈ (𝔼‘𝑛)((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ ∃𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
2725, 26bitr3i 280 . . . . . . . . 9 (((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ ∃𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
28272exbii 1850 . . . . . . . 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 278 . . . . . . 7 (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)) ↔ ∃𝑛𝑝𝑞(𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
30 anass 472 . . . . . . . . . 10 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ (𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))))
31 anass 472 . . . . . . . . . . 11 ((((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) ∧ 𝑥 = (𝑝Line𝑞)) ↔ ((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))))
32 simplrl 776 . . . . . . . . . . . . . 14 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → 𝑛 ∈ ℕ)
33 simplrr 777 . . . . . . . . . . . . . . 15 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → 𝑝 ∈ (𝔼‘𝑛))
34 simpll 766 . . . . . . . . . . . . . . 15 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → 𝑞 ∈ (𝔼‘𝑛))
35 simpr 488 . . . . . . . . . . . . . . 15 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → 𝑝𝑞)
3633, 34, 353jca 1125 . . . . . . . . . . . . . 14 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞))
3732, 36jca 515 . . . . . . . . . . . . 13 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → (𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)))
38 simpr2 1192 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → 𝑞 ∈ (𝔼‘𝑛))
39 simpl 486 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → 𝑛 ∈ ℕ)
40 simpr1 1191 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → 𝑝 ∈ (𝔼‘𝑛))
4138, 39, 40jca32 519 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → (𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))))
42 simpr3 1193 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → 𝑝𝑞)
4341, 42jca 515 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → ((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞))
4437, 43impbii 212 . . . . . . . . . . . 12 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) ↔ (𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)))
4544anbi1i 626 . . . . . . . . . . 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 280 . . . . . . . . . 10 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) ∧ 𝑥 = (𝑝Line𝑞)))
4730, 46bitr3i 280 . . . . . . . . 9 ((𝑞 ∈ (𝔼‘𝑛) ∧ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞)))) ↔ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) ∧ 𝑥 = (𝑝Line𝑞)))
48 fvline 34021 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → (𝑝Line𝑞) = {𝑥𝑥 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})
5149, 50ax-mp 5 . . . . . . . . . . . . 13 [⟨𝑝, 𝑞⟩] Colinear = {𝑥 ∣ ⟨𝑝, 𝑞 Colinear 𝑥}
52 vex 3413 . . . . . . . . . . . . . . 15 𝑥 ∈ V
5349, 52brcnv 5727 . . . . . . . . . . . . . 14 (⟨𝑝, 𝑞 Colinear 𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩)
5453abbii 2823 . . . . . . . . . . . . 13 {𝑥 ∣ ⟨𝑝, 𝑞 Colinear 𝑥} = {𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩}
5551, 54eqtri 2781 . . . . . . . . . . . 12 [⟨𝑝, 𝑞⟩] Colinear = {𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩}
5648, 55eqtr4di 2811 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → (𝑝Line𝑞) = [⟨𝑝, 𝑞⟩] Colinear )
5756eqeq2d 2769 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → (𝑥 = (𝑝Line𝑞) ↔ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear ))
5857pm5.32i 578 . . . . . . . . 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 472 . . . . . . . . 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 301 . . . . . . . 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 1851 . . . . . . 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 307 . . . . . 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 278 . . . . 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 278 . . . 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 278 . . 3 (𝑥 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)))
669, 13, 65vtoclbg 3489 . 2 (𝐴 ∈ V → (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞))))
671, 8, 66pm5.21nii 383 1 (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (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
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