| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | elex 3501 | . 2
⊢ (𝐴 ∈ LinesEE → 𝐴 ∈ V) | 
| 2 |  | ovex 7464 | . . . . . . 7
⊢ (𝑝Line𝑞) ∈ V | 
| 3 |  | eleq1 2829 | . . . . . . 7
⊢ (𝐴 = (𝑝Line𝑞) → (𝐴 ∈ V ↔ (𝑝Line𝑞) ∈ V)) | 
| 4 | 2, 3 | mpbiri 258 | . . . . . 6
⊢ (𝐴 = (𝑝Line𝑞) → 𝐴 ∈ V) | 
| 5 | 4 | adantl 481 | . . . . 5
⊢ ((𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V) | 
| 6 | 5 | rexlimivw 3151 | . . . 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 3201 | . 2
⊢
(∃𝑛 ∈
ℕ ∃𝑝 ∈
(𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)) → 𝐴 ∈ V) | 
| 9 |  | eleq1 2829 | . . 3
⊢ (𝑥 = 𝐴 → (𝑥 ∈ LinesEE ↔ 𝐴 ∈ LinesEE)) | 
| 10 |  | eqeq1 2741 | . . . . . 6
⊢ (𝑥 = 𝐴 → (𝑥 = (𝑝Line𝑞) ↔ 𝐴 = (𝑝Line𝑞))) | 
| 11 | 10 | anbi2d 630 | . . . . 5
⊢ (𝑥 = 𝐴 → ((𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)) ↔ (𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)))) | 
| 12 | 11 | rexbidv 3179 | . . . 4
⊢ (𝑥 = 𝐴 → (∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞)) ↔ ∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)))) | 
| 13 | 12 | 2rexbidv 3222 | . . 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 36140 | . . . . . 6
⊢ LinesEE =
ran Line | 
| 15 |  | df-line2 36138 | . . . . . . 7
⊢ Line =
{〈〈𝑝, 𝑞〉, 𝑥〉 ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [〈𝑝, 𝑞〉]◡ Colinear )} | 
| 16 | 15 | rneqi 5948 | . . . . . 6
⊢ ran Line
= ran {〈〈𝑝, 𝑞〉, 𝑥〉 ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [〈𝑝, 𝑞〉]◡ Colinear )} | 
| 17 |  | rnoprab 7538 | . . . . . 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 2769 | . . . . 5
⊢ LinesEE =
{𝑥 ∣ ∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [〈𝑝, 𝑞〉]◡ Colinear )} | 
| 19 | 18 | eleq2i 2833 | . . . 4
⊢ (𝑥 ∈ LinesEE ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑝∃𝑞∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞) ∧ 𝑥 = [〈𝑝, 𝑞〉]◡ Colinear )}) | 
| 20 |  | abid 2718 | . . . . 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 3071 | . . . . . . 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 1849 | . . . . . 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 3196 | . . . . . . . 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 3191 | . . . . . . . . . 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 3071 | . . . . . . . . . 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 277 | . . . . . . . . 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 1849 | . . . . . . . 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 275 | . . . . . . 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 468 | . . . . . . . . . 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 468 | . . . . . . . . . . 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 777 | . . . . . . . . . . . . . 14
⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑛 ∈ ℕ) | 
| 33 |  | simplrr 778 | . . . . . . . . . . . . . . 15
⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑝 ∈ (𝔼‘𝑛)) | 
| 34 |  | simpll 767 | . . . . . . . . . . . . . . 15
⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑞 ∈ (𝔼‘𝑛)) | 
| 35 |  | simpr 484 | . . . . . . . . . . . . . . 15
⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → 𝑝 ≠ 𝑞) | 
| 36 | 33, 34, 35 | 3jca 1129 | . . . . . . . . . . . . . 14
⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) | 
| 37 | 32, 36 | jca 511 | . . . . . . . . . . . . 13
⊢ (((𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛))) ∧ 𝑝 ≠ 𝑞) → (𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞))) | 
| 38 |  | simpr2 1196 | . . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑞 ∈ (𝔼‘𝑛)) | 
| 39 |  | simpl 482 | . . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑛 ∈ ℕ) | 
| 40 |  | simpr1 1195 | . . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑝 ∈ (𝔼‘𝑛)) | 
| 41 | 38, 39, 40 | jca32 515 | . . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑞 ∈ (𝔼‘𝑛) ∧ (𝑛 ∈ ℕ ∧ 𝑝 ∈ (𝔼‘𝑛)))) | 
| 42 |  | simpr3 1197 | . . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → 𝑝 ≠ 𝑞) | 
| 43 | 41, 42 | jca 511 | . . . . . . . . . . . . 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 209 | . . . . . . . . . . . 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 277 | . . . . . . . . . 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 277 | . . . . . . . . 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 36145 | . . . . . . . . . . . 12
⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑝Line𝑞) = {𝑥 ∣ 𝑥 Colinear 〈𝑝, 𝑞〉}) | 
| 49 |  | opex 5469 | . . . . . . . . . . . . . 14
⊢
〈𝑝, 𝑞〉 ∈ V | 
| 50 |  | dfec2 8748 | . . . . . . . . . . . . . 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 3484 | . . . . . . . . . . . . . . 15
⊢ 𝑥 ∈ V | 
| 53 | 49, 52 | brcnv 5893 | . . . . . . . . . . . . . 14
⊢
(〈𝑝, 𝑞〉◡ Colinear 𝑥 ↔ 𝑥 Colinear 〈𝑝, 𝑞〉) | 
| 54 | 53 | abbii 2809 | . . . . . . . . . . . . 13
⊢ {𝑥 ∣ 〈𝑝, 𝑞〉◡ Colinear 𝑥} = {𝑥 ∣ 𝑥 Colinear 〈𝑝, 𝑞〉} | 
| 55 | 51, 54 | eqtri 2765 | . . . . . . . . . . . 12
⊢
[〈𝑝, 𝑞〉]◡ Colinear = {𝑥 ∣ 𝑥 Colinear 〈𝑝, 𝑞〉} | 
| 56 | 48, 55 | eqtr4di 2795 | . . . . . . . . . . 11
⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑝Line𝑞) = [〈𝑝, 𝑞〉]◡ Colinear ) | 
| 57 | 56 | eqeq2d 2748 | . . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ (𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑞)) → (𝑥 = (𝑝Line𝑞) ↔ 𝑥 = [〈𝑝, 𝑞〉]◡ Colinear )) | 
| 58 | 57 | pm5.32i 574 | . . . . . . . . 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 468 | . . . . . . . . 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 298 | . . . . . . . 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 1850 | . . . . . . 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 304 | . . . . . 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 275 | . . . . 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 275 | . . . 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 275 | . . 3
⊢ (𝑥 ∈ LinesEE ↔
∃𝑛 ∈ ℕ
∃𝑝 ∈
(𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝑥 = (𝑝Line𝑞))) | 
| 66 | 9, 13, 65 | vtoclbg 3557 | . 2
⊢ (𝐴 ∈ V → (𝐴 ∈ LinesEE ↔
∃𝑛 ∈ ℕ
∃𝑝 ∈
(𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)))) | 
| 67 | 1, 8, 66 | pm5.21nii 378 | 1
⊢ (𝐴 ∈ LinesEE ↔
∃𝑛 ∈ ℕ
∃𝑝 ∈
(𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞))) |