Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ellines Structured version   Visualization version   GIF version

Theorem ellines 35748
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 3490 . 2 (𝐴 ∈ LinesEE → 𝐴 ∈ V)
2 ovex 7453 . . . . . . 7 (𝑝Line𝑞) ∈ V
3 eleq1 2817 . . . . . . 7 (𝐴 = (𝑝Line𝑞) → (𝐴 ∈ V ↔ (𝑝Line𝑞) ∈ V))
42, 3mpbiri 258 . . . . . 6 (𝐴 = (𝑝Line𝑞) → 𝐴 ∈ V)
54adantl 481 . . . . 5 ((𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
65rexlimivw 3148 . . . 4 (∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
76a1i 11 . . 3 ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V))
87rexlimivv 3196 . 2 (∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V)
9 eleq1 2817 . . 3 (𝑥 = 𝐴 → (𝑥 ∈ LinesEE ↔ 𝐴 ∈ LinesEE))
10 eqeq1 2732 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = (𝑝Line𝑞) ↔ 𝐴 = (𝑝Line𝑞)))
1110anbi2d 629 . . . . 5 (𝑥 = 𝐴 → ((𝑝𝑞𝑥 = (𝑝Line𝑞)) ↔ (𝑝𝑞𝐴 = (𝑝Line𝑞))))
1211rexbidv 3175 . . . 4 (𝑥 = 𝐴 → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞)) ↔ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞))))
13122rexbidv 3216 . . 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 35735 . . . . . 6 LinesEE = ran Line
15 df-line2 35733 . . . . . . 7 Line = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )}
1615rneqi 5939 . . . . . 6 ran Line = ran {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )}
17 rnoprab 7524 . . . . . 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 2760 . . . . 5 LinesEE = {𝑥 ∣ ∃𝑝𝑞𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )}
1918eleq2i 2821 . . . 4 (𝑥 ∈ LinesEE ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑝𝑞𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞) ∧ 𝑥 = [⟨𝑝, 𝑞⟩] Colinear )})
20 abid 2709 . . . . 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 3068 . . . . . . 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 1844 . . . . . 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 2155 . . . . . . 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 3192 . . . . . . . 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 3187 . . . . . . . . . 10 (∃𝑞 ∈ (𝔼‘𝑛)((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ (𝑝𝑞𝑥 = (𝑝Line𝑞))) ↔ ((𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)) ∧ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝑥 = (𝑝Line𝑞))))
26 df-rex 3068 . . . . . . . . . 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 1844 . . . . . . . 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 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 484 . . . . . . . . . . . . . . 15 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) → 𝑝𝑞)
3633, 34, 353jca 1126 . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . 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 1192 . . . . . . . . . . . . . . 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 1194 . . . . . . . . . . . . . 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 208 . . . . . . . . . . . 12 (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝𝑞) ↔ (𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)))
4544anbi1i 623 . . . . . . . . . . 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 35740 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → (𝑝Line𝑞) = {𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩})
49 opex 5466 . . . . . . . . . . . . . 14 𝑝, 𝑞⟩ ∈ V
50 dfec2 8727 . . . . . . . . . . . . . 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 3475 . . . . . . . . . . . . . . 15 𝑥 ∈ V
5349, 52brcnv 5885 . . . . . . . . . . . . . 14 (⟨𝑝, 𝑞 Colinear 𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩)
5453abbii 2798 . . . . . . . . . . . . 13 {𝑥 ∣ ⟨𝑝, 𝑞 Colinear 𝑥} = {𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩}
5551, 54eqtri 2756 . . . . . . . . . . . 12 [⟨𝑝, 𝑞⟩] Colinear = {𝑥𝑥 Colinear ⟨𝑝, 𝑞⟩}
5648, 55eqtr4di 2786 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝𝑞)) → (𝑝Line𝑞) = [⟨𝑝, 𝑞⟩] Colinear )
5756eqeq2d 2739 . . . . . . . . . 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 1845 . . . . . . 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 3542 . 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 205  wa 395  w3a 1085   = wceq 1534  wex 1774  wcel 2099  {cab 2705  wne 2937  wrex 3067  Vcvv 3471  cop 4635   class class class wbr 5148  ccnv 5677  ran crn 5679  cfv 6548  (class class class)co 7420  {coprab 7421  [cec 8722  cn 12242  𝔼cee 28698   Colinear ccolin 35633  Linecline2 35730  LinesEEclines2 35732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11194  ax-1cn 11196  ax-addcl 11198
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-om 7871  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-ec 8726  df-nn 12243  df-colinear 35635  df-line2 35733  df-lines2 35735
This theorem is referenced by:  linethru  35749  hilbert1.1  35750
  Copyright terms: Public domain W3C validator