Step | Hyp | Ref
| Expression |
1 | | elxp2 5610 |
. . 3
⊢ (𝐴 ∈ (N ×
N) ↔ ∃𝑎 ∈ N ∃𝑏 ∈ N 𝐴 = 〈𝑎, 𝑏〉) |
2 | | pion 10624 |
. . . . . . . . 9
⊢ (𝑏 ∈ N →
𝑏 ∈
On) |
3 | | suceloni 7651 |
. . . . . . . . 9
⊢ (𝑏 ∈ On → suc 𝑏 ∈ On) |
4 | 2, 3 | syl 17 |
. . . . . . . 8
⊢ (𝑏 ∈ N →
suc 𝑏 ∈
On) |
5 | | vex 3435 |
. . . . . . . . 9
⊢ 𝑏 ∈ V |
6 | 5 | sucid 6340 |
. . . . . . . 8
⊢ 𝑏 ∈ suc 𝑏 |
7 | | eleq2 2827 |
. . . . . . . . 9
⊢ (𝑦 = suc 𝑏 → (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ suc 𝑏)) |
8 | 7 | rspcev 3561 |
. . . . . . . 8
⊢ ((suc
𝑏 ∈ On ∧ 𝑏 ∈ suc 𝑏) → ∃𝑦 ∈ On 𝑏 ∈ 𝑦) |
9 | 4, 6, 8 | sylancl 586 |
. . . . . . 7
⊢ (𝑏 ∈ N →
∃𝑦 ∈ On 𝑏 ∈ 𝑦) |
10 | 9 | adantl 482 |
. . . . . 6
⊢ ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ ∃𝑦 ∈ On
𝑏 ∈ 𝑦) |
11 | | elequ2 2121 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑚 → (𝑏 ∈ 𝑦 ↔ 𝑏 ∈ 𝑚)) |
12 | 11 | imbi1d 342 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑚 → ((𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉) ↔ (𝑏 ∈ 𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉))) |
13 | 12 | 2ralbidv 3120 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑚 → (∀𝑎 ∈ N ∀𝑏 ∈ N (𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉) ↔ ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑏 ∈ 𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉))) |
14 | | opeq1 4806 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑐 = 𝑎 → 〈𝑐, 𝑑〉 = 〈𝑎, 𝑑〉) |
15 | 14 | breq2d 5087 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 = 𝑎 → (𝑥 ~Q 〈𝑐, 𝑑〉 ↔ 𝑥 ~Q 〈𝑎, 𝑑〉)) |
16 | 15 | rexbidv 3225 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 = 𝑎 → (∃𝑥 ∈ Q 𝑥 ~Q 〈𝑐, 𝑑〉 ↔ ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑑〉)) |
17 | 16 | imbi2d 341 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐 = 𝑎 → ((𝑑 ∈ 𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑐, 𝑑〉) ↔ (𝑑 ∈ 𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑑〉))) |
18 | | elequ1 2113 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑑 = 𝑏 → (𝑑 ∈ 𝑚 ↔ 𝑏 ∈ 𝑚)) |
19 | | opeq2 4807 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 = 𝑏 → 〈𝑎, 𝑑〉 = 〈𝑎, 𝑏〉) |
20 | 19 | breq2d 5087 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑑 = 𝑏 → (𝑥 ~Q 〈𝑎, 𝑑〉 ↔ 𝑥 ~Q 〈𝑎, 𝑏〉)) |
21 | 20 | rexbidv 3225 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑑 = 𝑏 → (∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑑〉 ↔ ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉)) |
22 | 18, 21 | imbi12d 345 |
. . . . . . . . . . . . . . . 16
⊢ (𝑑 = 𝑏 → ((𝑑 ∈ 𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑑〉) ↔ (𝑏 ∈ 𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉))) |
23 | 17, 22 | cbvral2vw 3395 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑐 ∈
N ∀𝑑
∈ N (𝑑
∈ 𝑚 →
∃𝑥 ∈
Q 𝑥
~Q 〈𝑐, 𝑑〉) ↔ ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑏 ∈ 𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉)) |
24 | 23 | ralbii 3092 |
. . . . . . . . . . . . . 14
⊢
(∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) ↔ ∀𝑚 ∈ 𝑦 ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑏 ∈ 𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉)) |
25 | | rexnal 3168 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∃𝑧 ∈
(N × N) ¬ (〈𝑎, 𝑏〉 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N 𝑏) ↔ ¬ ∀𝑧 ∈ (N ×
N)(〈𝑎,
𝑏〉
~Q 𝑧 → ¬ (2nd ‘𝑧) <N
𝑏)) |
26 | | pm4.63 398 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (¬
(〈𝑎, 𝑏〉
~Q 𝑧 → ¬ (2nd ‘𝑧) <N
𝑏) ↔ (〈𝑎, 𝑏〉 ~Q 𝑧 ∧ (2nd
‘𝑧)
<N 𝑏)) |
27 | | xp2nd 7855 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑧 ∈ (N ×
N) → (2nd ‘𝑧) ∈ N) |
28 | | ltpiord 10632 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((2nd ‘𝑧) ∈ N ∧ 𝑏 ∈ N) →
((2nd ‘𝑧)
<N 𝑏 ↔ (2nd ‘𝑧) ∈ 𝑏)) |
29 | 28 | ancoms 459 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑏 ∈ N ∧
(2nd ‘𝑧)
∈ N) → ((2nd ‘𝑧) <N 𝑏 ↔ (2nd
‘𝑧) ∈ 𝑏)) |
30 | 27, 29 | sylan2 593 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑏 ∈ N ∧
𝑧 ∈ (N
× N)) → ((2nd ‘𝑧) <N 𝑏 ↔ (2nd
‘𝑧) ∈ 𝑏)) |
31 | 30 | adantll 711 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑎 ∈ N ∧
𝑏 ∈ N)
∧ 𝑧 ∈
(N × N)) → ((2nd ‘𝑧) <N
𝑏 ↔ (2nd
‘𝑧) ∈ 𝑏)) |
32 | 31 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑎 ∈ N ∧
𝑏 ∈ N)
∧ 𝑧 ∈
(N × N)) → ((〈𝑎, 𝑏〉 ~Q 𝑧 ∧ (2nd
‘𝑧)
<N 𝑏) ↔ (〈𝑎, 𝑏〉 ~Q 𝑧 ∧ (2nd
‘𝑧) ∈ 𝑏))) |
33 | 26, 32 | bitrid 282 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑎 ∈ N ∧
𝑏 ∈ N)
∧ 𝑧 ∈
(N × N)) → (¬ (〈𝑎, 𝑏〉 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N 𝑏) ↔ (〈𝑎, 𝑏〉 ~Q 𝑧 ∧ (2nd
‘𝑧) ∈ 𝑏))) |
34 | 33 | rexbidva 3224 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ (∃𝑧 ∈
(N × N) ¬ (〈𝑎, 𝑏〉 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N 𝑏) ↔ ∃𝑧 ∈ (N ×
N)(〈𝑎,
𝑏〉
~Q 𝑧 ∧ (2nd ‘𝑧) ∈ 𝑏))) |
35 | 25, 34 | bitr3id 285 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ (¬ ∀𝑧
∈ (N × N)(〈𝑎, 𝑏〉 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N 𝑏) ↔ ∃𝑧 ∈ (N ×
N)(〈𝑎,
𝑏〉
~Q 𝑧 ∧ (2nd ‘𝑧) ∈ 𝑏))) |
36 | | xp1st 7854 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑧 ∈ (N ×
N) → (1st ‘𝑧) ∈ N) |
37 | | elequ2 2121 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑚 = 𝑏 → (𝑑 ∈ 𝑚 ↔ 𝑑 ∈ 𝑏)) |
38 | 37 | imbi1d 342 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑚 = 𝑏 → ((𝑑 ∈ 𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑐, 𝑑〉) ↔ (𝑑 ∈ 𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑐, 𝑑〉))) |
39 | 38 | 2ralbidv 3120 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑚 = 𝑏 → (∀𝑐 ∈ N ∀𝑑 ∈ N (𝑑 ∈ 𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑐, 𝑑〉) ↔ ∀𝑐 ∈ N ∀𝑑 ∈ N (𝑑 ∈ 𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑐, 𝑑〉))) |
40 | 39 | rspccv 3558 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) → (𝑏 ∈ 𝑦 → ∀𝑐 ∈ N ∀𝑑 ∈ N (𝑑 ∈ 𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑐, 𝑑〉))) |
41 | | opeq1 4806 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑐 = (1st ‘𝑧) → 〈𝑐, 𝑑〉 = 〈(1st ‘𝑧), 𝑑〉) |
42 | 41 | breq2d 5087 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑐 = (1st ‘𝑧) → (𝑥 ~Q 〈𝑐, 𝑑〉 ↔ 𝑥 ~Q
〈(1st ‘𝑧), 𝑑〉)) |
43 | 42 | rexbidv 3225 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑐 = (1st ‘𝑧) → (∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉 ↔ ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), 𝑑〉)) |
44 | 43 | imbi2d 341 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑐 = (1st ‘𝑧) → ((𝑑 ∈ 𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑐, 𝑑〉) ↔ (𝑑 ∈ 𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), 𝑑〉))) |
45 | 44 | ralbidv 3119 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑐 = (1st ‘𝑧) → (∀𝑑 ∈ N (𝑑 ∈ 𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑐, 𝑑〉) ↔ ∀𝑑 ∈ N (𝑑 ∈ 𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), 𝑑〉))) |
46 | 45 | rspccv 3558 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
(∀𝑐 ∈
N ∀𝑑
∈ N (𝑑
∈ 𝑏 →
∃𝑥 ∈
Q 𝑥
~Q 〈𝑐, 𝑑〉) → ((1st ‘𝑧) ∈ N →
∀𝑑 ∈
N (𝑑 ∈
𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), 𝑑〉))) |
47 | | eleq1 2826 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑑 = (2nd ‘𝑧) → (𝑑 ∈ 𝑏 ↔ (2nd ‘𝑧) ∈ 𝑏)) |
48 | | opeq2 4807 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑑 = (2nd ‘𝑧) → 〈(1st
‘𝑧), 𝑑〉 = 〈(1st
‘𝑧), (2nd
‘𝑧)〉) |
49 | 48 | breq2d 5087 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑑 = (2nd ‘𝑧) → (𝑥 ~Q
〈(1st ‘𝑧), 𝑑〉 ↔ 𝑥 ~Q
〈(1st ‘𝑧), (2nd ‘𝑧)〉)) |
50 | 49 | rexbidv 3225 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑑 = (2nd ‘𝑧) → (∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), 𝑑〉 ↔ ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), (2nd ‘𝑧)〉)) |
51 | 47, 50 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑑 = (2nd ‘𝑧) → ((𝑑 ∈ 𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), 𝑑〉) ↔ ((2nd ‘𝑧) ∈ 𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), (2nd ‘𝑧)〉))) |
52 | 51 | rspccv 3558 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
(∀𝑑 ∈
N (𝑑 ∈
𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), 𝑑〉) → ((2nd ‘𝑧) ∈ N →
((2nd ‘𝑧)
∈ 𝑏 →
∃𝑥 ∈
Q 𝑥
~Q 〈(1st ‘𝑧), (2nd ‘𝑧)〉))) |
53 | 46, 52 | syl6 35 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(∀𝑐 ∈
N ∀𝑑
∈ N (𝑑
∈ 𝑏 →
∃𝑥 ∈
Q 𝑥
~Q 〈𝑐, 𝑑〉) → ((1st ‘𝑧) ∈ N →
((2nd ‘𝑧)
∈ N → ((2nd ‘𝑧) ∈ 𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), (2nd ‘𝑧)〉)))) |
54 | 40, 53 | syl6 35 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) → (𝑏 ∈ 𝑦 → ((1st ‘𝑧) ∈ N →
((2nd ‘𝑧)
∈ N → ((2nd ‘𝑧) ∈ 𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), (2nd ‘𝑧)〉))))) |
55 | 54 | imp 407 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) ∧ 𝑏 ∈ 𝑦) → ((1st ‘𝑧) ∈ N →
((2nd ‘𝑧)
∈ N → ((2nd ‘𝑧) ∈ 𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), (2nd ‘𝑧)〉)))) |
56 | 36, 55 | syl5 34 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) ∧ 𝑏 ∈ 𝑦) → (𝑧 ∈ (N ×
N) → ((2nd ‘𝑧) ∈ N →
((2nd ‘𝑧)
∈ 𝑏 →
∃𝑥 ∈
Q 𝑥
~Q 〈(1st ‘𝑧), (2nd ‘𝑧)〉)))) |
57 | 27, 56 | mpdi 45 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) ∧ 𝑏 ∈ 𝑦) → (𝑧 ∈ (N ×
N) → ((2nd ‘𝑧) ∈ 𝑏 → ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), (2nd ‘𝑧)〉))) |
58 | 57 | 3imp 1110 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) ∧ 𝑏 ∈ 𝑦) ∧ 𝑧 ∈ (N ×
N) ∧ (2nd ‘𝑧) ∈ 𝑏) → ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), (2nd ‘𝑧)〉) |
59 | | 1st2nd2 7861 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑧 ∈ (N ×
N) → 𝑧 =
〈(1st ‘𝑧), (2nd ‘𝑧)〉) |
60 | 59 | breq2d 5087 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑧 ∈ (N ×
N) → (𝑥
~Q 𝑧 ↔ 𝑥 ~Q
〈(1st ‘𝑧), (2nd ‘𝑧)〉)) |
61 | 60 | rexbidv 3225 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧 ∈ (N ×
N) → (∃𝑥 ∈ Q 𝑥 ~Q 𝑧 ↔ ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), (2nd ‘𝑧)〉)) |
62 | 61 | 3ad2ant2 1133 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) ∧ 𝑏 ∈ 𝑦) ∧ 𝑧 ∈ (N ×
N) ∧ (2nd ‘𝑧) ∈ 𝑏) → (∃𝑥 ∈ Q 𝑥 ~Q 𝑧 ↔ ∃𝑥 ∈ Q 𝑥 ~Q
〈(1st ‘𝑧), (2nd ‘𝑧)〉)) |
63 | 58, 62 | mpbird 256 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) ∧ 𝑏 ∈ 𝑦) ∧ 𝑧 ∈ (N ×
N) ∧ (2nd ‘𝑧) ∈ 𝑏) → ∃𝑥 ∈ Q 𝑥 ~Q 𝑧) |
64 | | enqer 10666 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
~Q Er (N ×
N) |
65 | 64 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((〈𝑎, 𝑏〉
~Q 𝑧 ∧ 𝑥 ~Q 𝑧) →
~Q Er (N ×
N)) |
66 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((〈𝑎, 𝑏〉
~Q 𝑧 ∧ 𝑥 ~Q 𝑧) → 𝑥 ~Q 𝑧) |
67 | | simpl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((〈𝑎, 𝑏〉
~Q 𝑧 ∧ 𝑥 ~Q 𝑧) → 〈𝑎, 𝑏〉 ~Q 𝑧) |
68 | 65, 66, 67 | ertr4d 8506 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((〈𝑎, 𝑏〉
~Q 𝑧 ∧ 𝑥 ~Q 𝑧) → 𝑥 ~Q 〈𝑎, 𝑏〉) |
69 | 68 | ex 413 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(〈𝑎, 𝑏〉
~Q 𝑧 → (𝑥 ~Q 𝑧 → 𝑥 ~Q 〈𝑎, 𝑏〉)) |
70 | 69 | reximdv 3201 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(〈𝑎, 𝑏〉
~Q 𝑧 → (∃𝑥 ∈ Q 𝑥 ~Q 𝑧 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉)) |
71 | 63, 70 | syl5com 31 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) ∧ 𝑏 ∈ 𝑦) ∧ 𝑧 ∈ (N ×
N) ∧ (2nd ‘𝑧) ∈ 𝑏) → (〈𝑎, 𝑏〉 ~Q 𝑧 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉)) |
72 | 71 | 3expia 1120 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) ∧ 𝑏 ∈ 𝑦) ∧ 𝑧 ∈ (N ×
N)) → ((2nd ‘𝑧) ∈ 𝑏 → (〈𝑎, 𝑏〉 ~Q 𝑧 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉))) |
73 | 72 | impcomd 412 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) ∧ 𝑏 ∈ 𝑦) ∧ 𝑧 ∈ (N ×
N)) → ((〈𝑎, 𝑏〉 ~Q 𝑧 ∧ (2nd
‘𝑧) ∈ 𝑏) → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉)) |
74 | 73 | rexlimdva 3212 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) ∧ 𝑏 ∈ 𝑦) → (∃𝑧 ∈ (N ×
N)(〈𝑎,
𝑏〉
~Q 𝑧 ∧ (2nd ‘𝑧) ∈ 𝑏) → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉)) |
75 | 74 | ex 413 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) → (𝑏 ∈ 𝑦 → (∃𝑧 ∈ (N ×
N)(〈𝑎,
𝑏〉
~Q 𝑧 ∧ (2nd ‘𝑧) ∈ 𝑏) → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉))) |
76 | 75 | com3r 87 |
. . . . . . . . . . . . . . . . . 18
⊢
(∃𝑧 ∈
(N × N)(〈𝑎, 𝑏〉 ~Q 𝑧 ∧ (2nd
‘𝑧) ∈ 𝑏) → (∀𝑚 ∈ 𝑦 ∀𝑐 ∈ N ∀𝑑 ∈ N (𝑑 ∈ 𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑐, 𝑑〉) → (𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉))) |
77 | 35, 76 | syl6bi 252 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ (¬ ∀𝑧
∈ (N × N)(〈𝑎, 𝑏〉 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N 𝑏) → (∀𝑚 ∈ 𝑦 ∀𝑐 ∈ N ∀𝑑 ∈ N (𝑑 ∈ 𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑐, 𝑑〉) → (𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉)))) |
78 | 77 | com13 88 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) → (¬ ∀𝑧 ∈ (N ×
N)(〈𝑎,
𝑏〉
~Q 𝑧 → ¬ (2nd ‘𝑧) <N
𝑏) → ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ (𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉)))) |
79 | | mulcompi 10641 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑎
·N 𝑏) = (𝑏 ·N 𝑎) |
80 | | enqbreq 10664 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑎 ∈ N ∧
𝑏 ∈ N)
∧ (𝑎 ∈
N ∧ 𝑏
∈ N)) → (〈𝑎, 𝑏〉 ~Q
〈𝑎, 𝑏〉 ↔ (𝑎 ·N 𝑏) = (𝑏 ·N 𝑎))) |
81 | 80 | anidms 567 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ (〈𝑎, 𝑏〉
~Q 〈𝑎, 𝑏〉 ↔ (𝑎 ·N 𝑏) = (𝑏 ·N 𝑎))) |
82 | 79, 81 | mpbiri 257 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ 〈𝑎, 𝑏〉
~Q 〈𝑎, 𝑏〉) |
83 | | opelxpi 5623 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ 〈𝑎, 𝑏〉 ∈ (N
× N)) |
84 | | breq1 5078 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = 〈𝑎, 𝑏〉 → (𝑦 ~Q 𝑧 ↔ 〈𝑎, 𝑏〉 ~Q 𝑧)) |
85 | | vex 3435 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ 𝑎 ∈ V |
86 | 85, 5 | op2ndd 7833 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 〈𝑎, 𝑏〉 → (2nd ‘𝑦) = 𝑏) |
87 | 86 | breq2d 5087 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 = 〈𝑎, 𝑏〉 → ((2nd ‘𝑧) <N
(2nd ‘𝑦)
↔ (2nd ‘𝑧) <N 𝑏)) |
88 | 87 | notbid 318 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = 〈𝑎, 𝑏〉 → (¬ (2nd
‘𝑧)
<N (2nd ‘𝑦) ↔ ¬ (2nd ‘𝑧) <N
𝑏)) |
89 | 84, 88 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = 〈𝑎, 𝑏〉 → ((𝑦 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N (2nd ‘𝑦)) ↔ (〈𝑎, 𝑏〉 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N 𝑏))) |
90 | 89 | ralbidv 3119 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = 〈𝑎, 𝑏〉 → (∀𝑧 ∈ (N ×
N)(𝑦
~Q 𝑧 → ¬ (2nd ‘𝑧) <N
(2nd ‘𝑦))
↔ ∀𝑧 ∈
(N × N)(〈𝑎, 𝑏〉 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N 𝑏))) |
91 | | df-nq 10657 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Q = {𝑦
∈ (N × N) ∣ ∀𝑧 ∈ (N ×
N)(𝑦
~Q 𝑧 → ¬ (2nd ‘𝑧) <N
(2nd ‘𝑦))} |
92 | 90, 91 | elrab2 3628 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(〈𝑎, 𝑏〉 ∈ Q
↔ (〈𝑎, 𝑏〉 ∈ (N
× N) ∧ ∀𝑧 ∈ (N ×
N)(〈𝑎,
𝑏〉
~Q 𝑧 → ¬ (2nd ‘𝑧) <N
𝑏))) |
93 | 92 | simplbi2 501 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(〈𝑎, 𝑏〉 ∈ (N
× N) → (∀𝑧 ∈ (N ×
N)(〈𝑎,
𝑏〉
~Q 𝑧 → ¬ (2nd ‘𝑧) <N
𝑏) → 〈𝑎, 𝑏〉 ∈
Q)) |
94 | 83, 93 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ (∀𝑧 ∈
(N × N)(〈𝑎, 𝑏〉 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N 𝑏) → 〈𝑎, 𝑏〉 ∈
Q)) |
95 | | breq1 5078 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 〈𝑎, 𝑏〉 → (𝑥 ~Q 〈𝑎, 𝑏〉 ↔ 〈𝑎, 𝑏〉 ~Q
〈𝑎, 𝑏〉)) |
96 | 95 | rspcev 3561 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((〈𝑎, 𝑏〉 ∈ Q
∧ 〈𝑎, 𝑏〉
~Q 〈𝑎, 𝑏〉) → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉) |
97 | 96 | expcom 414 |
. . . . . . . . . . . . . . . . . . 19
⊢
(〈𝑎, 𝑏〉
~Q 〈𝑎, 𝑏〉 → (〈𝑎, 𝑏〉 ∈ Q →
∃𝑥 ∈
Q 𝑥
~Q 〈𝑎, 𝑏〉)) |
98 | 82, 94, 97 | sylsyld 61 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ (∀𝑧 ∈
(N × N)(〈𝑎, 𝑏〉 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N 𝑏) → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉)) |
99 | 98 | com12 32 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑧 ∈
(N × N)(〈𝑎, 𝑏〉 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N 𝑏) → ((𝑎 ∈ N ∧ 𝑏 ∈ N) →
∃𝑥 ∈
Q 𝑥
~Q 〈𝑎, 𝑏〉)) |
100 | 99 | a1dd 50 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑧 ∈
(N × N)(〈𝑎, 𝑏〉 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N 𝑏) → ((𝑎 ∈ N ∧ 𝑏 ∈ N) →
(𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉))) |
101 | 78, 100 | pm2.61d2 181 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) → ((𝑎 ∈ N ∧ 𝑏 ∈ N) →
(𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉))) |
102 | 101 | ralrimivv 3110 |
. . . . . . . . . . . . . 14
⊢
(∀𝑚 ∈
𝑦 ∀𝑐 ∈ N
∀𝑑 ∈
N (𝑑 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑐, 𝑑〉) → ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉)) |
103 | 24, 102 | sylbir 234 |
. . . . . . . . . . . . 13
⊢
(∀𝑚 ∈
𝑦 ∀𝑎 ∈ N
∀𝑏 ∈
N (𝑏 ∈
𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉) → ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉)) |
104 | 103 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ On → (∀𝑚 ∈ 𝑦 ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑏 ∈ 𝑚 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉) → ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉))) |
105 | 13, 104 | tfis2 7695 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ On → ∀𝑎 ∈ N
∀𝑏 ∈
N (𝑏 ∈
𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉)) |
106 | | rsp 3131 |
. . . . . . . . . . 11
⊢
(∀𝑎 ∈
N ∀𝑏
∈ N (𝑏
∈ 𝑦 →
∃𝑥 ∈
Q 𝑥
~Q 〈𝑎, 𝑏〉) → (𝑎 ∈ N → ∀𝑏 ∈ N (𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉))) |
107 | 105, 106 | syl 17 |
. . . . . . . . . 10
⊢ (𝑦 ∈ On → (𝑎 ∈ N →
∀𝑏 ∈
N (𝑏 ∈
𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉))) |
108 | | rsp 3131 |
. . . . . . . . . 10
⊢
(∀𝑏 ∈
N (𝑏 ∈
𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉) → (𝑏 ∈ N → (𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉))) |
109 | 107, 108 | syl6 35 |
. . . . . . . . 9
⊢ (𝑦 ∈ On → (𝑎 ∈ N →
(𝑏 ∈ N
→ (𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉)))) |
110 | 109 | impd 411 |
. . . . . . . 8
⊢ (𝑦 ∈ On → ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ (𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉))) |
111 | 110 | com12 32 |
. . . . . . 7
⊢ ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ (𝑦 ∈ On →
(𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉))) |
112 | 111 | rexlimdv 3211 |
. . . . . 6
⊢ ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ (∃𝑦 ∈ On
𝑏 ∈ 𝑦 → ∃𝑥 ∈ Q 𝑥 ~Q 〈𝑎, 𝑏〉)) |
113 | 10, 112 | mpd 15 |
. . . . 5
⊢ ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ ∃𝑥 ∈
Q 𝑥
~Q 〈𝑎, 𝑏〉) |
114 | | breq2 5079 |
. . . . . 6
⊢ (𝐴 = 〈𝑎, 𝑏〉 → (𝑥 ~Q 𝐴 ↔ 𝑥 ~Q 〈𝑎, 𝑏〉)) |
115 | 114 | rexbidv 3225 |
. . . . 5
⊢ (𝐴 = 〈𝑎, 𝑏〉 → (∃𝑥 ∈ Q 𝑥 ~Q 𝐴 ↔ ∃𝑥 ∈ Q 𝑥 ~Q
〈𝑎, 𝑏〉)) |
116 | 113, 115 | syl5ibrcom 246 |
. . . 4
⊢ ((𝑎 ∈ N ∧
𝑏 ∈ N)
→ (𝐴 = 〈𝑎, 𝑏〉 → ∃𝑥 ∈ Q 𝑥 ~Q 𝐴)) |
117 | 116 | rexlimivv 3220 |
. . 3
⊢
(∃𝑎 ∈
N ∃𝑏
∈ N 𝐴 =
〈𝑎, 𝑏〉 → ∃𝑥 ∈ Q 𝑥 ~Q 𝐴) |
118 | 1, 117 | sylbi 216 |
. 2
⊢ (𝐴 ∈ (N ×
N) → ∃𝑥 ∈ Q 𝑥 ~Q 𝐴) |
119 | | breq2 5079 |
. . . . . 6
⊢ (𝑎 = 𝐴 → (𝑥 ~Q 𝑎 ↔ 𝑥 ~Q 𝐴)) |
120 | | breq2 5079 |
. . . . . 6
⊢ (𝑎 = 𝐴 → (𝑦 ~Q 𝑎 ↔ 𝑦 ~Q 𝐴)) |
121 | 119, 120 | anbi12d 631 |
. . . . 5
⊢ (𝑎 = 𝐴 → ((𝑥 ~Q 𝑎 ∧ 𝑦 ~Q 𝑎) ↔ (𝑥 ~Q 𝐴 ∧ 𝑦 ~Q 𝐴))) |
122 | 121 | imbi1d 342 |
. . . 4
⊢ (𝑎 = 𝐴 → (((𝑥 ~Q 𝑎 ∧ 𝑦 ~Q 𝑎) → 𝑥 = 𝑦) ↔ ((𝑥 ~Q 𝐴 ∧ 𝑦 ~Q 𝐴) → 𝑥 = 𝑦))) |
123 | 122 | 2ralbidv 3120 |
. . 3
⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ Q ∀𝑦 ∈ Q ((𝑥 ~Q
𝑎 ∧ 𝑦 ~Q 𝑎) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ Q ∀𝑦 ∈ Q ((𝑥 ~Q
𝐴 ∧ 𝑦 ~Q 𝐴) → 𝑥 = 𝑦))) |
124 | 64 | a1i 11 |
. . . . . 6
⊢ ((𝑥 ~Q
𝑎 ∧ 𝑦 ~Q 𝑎) →
~Q Er (N ×
N)) |
125 | | simpl 483 |
. . . . . 6
⊢ ((𝑥 ~Q
𝑎 ∧ 𝑦 ~Q 𝑎) → 𝑥 ~Q 𝑎) |
126 | | simpr 485 |
. . . . . 6
⊢ ((𝑥 ~Q
𝑎 ∧ 𝑦 ~Q 𝑎) → 𝑦 ~Q 𝑎) |
127 | 124, 125,
126 | ertr4d 8506 |
. . . . 5
⊢ ((𝑥 ~Q
𝑎 ∧ 𝑦 ~Q 𝑎) → 𝑥 ~Q 𝑦) |
128 | | mulcompi 10641 |
. . . . . . . . . . 11
⊢
((2nd ‘𝑥) ·N
(1st ‘𝑥))
= ((1st ‘𝑥) ·N
(2nd ‘𝑥)) |
129 | | elpqn 10670 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ Q →
𝑦 ∈ (N
× N)) |
130 | | breq1 5078 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑥 → (𝑦 ~Q 𝑧 ↔ 𝑥 ~Q 𝑧)) |
131 | | fveq2 6768 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑥 → (2nd ‘𝑦) = (2nd ‘𝑥)) |
132 | 131 | breq2d 5087 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑥 → ((2nd ‘𝑧) <N
(2nd ‘𝑦)
↔ (2nd ‘𝑧) <N
(2nd ‘𝑥))) |
133 | 132 | notbid 318 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑥 → (¬ (2nd ‘𝑧) <N
(2nd ‘𝑦)
↔ ¬ (2nd ‘𝑧) <N
(2nd ‘𝑥))) |
134 | 130, 133 | imbi12d 345 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = 𝑥 → ((𝑦 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N (2nd ‘𝑦)) ↔ (𝑥 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N (2nd ‘𝑥)))) |
135 | 134 | ralbidv 3119 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ (N ×
N)(𝑦
~Q 𝑧 → ¬ (2nd ‘𝑧) <N
(2nd ‘𝑦))
↔ ∀𝑧 ∈
(N × N)(𝑥 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N (2nd ‘𝑥)))) |
136 | 135, 91 | elrab2 3628 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ Q ↔
(𝑥 ∈ (N
× N) ∧ ∀𝑧 ∈ (N ×
N)(𝑥
~Q 𝑧 → ¬ (2nd ‘𝑧) <N
(2nd ‘𝑥)))) |
137 | 136 | simprbi 497 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ Q →
∀𝑧 ∈
(N × N)(𝑥 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N (2nd ‘𝑥))) |
138 | | breq2 5079 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝑦 → (𝑥 ~Q 𝑧 ↔ 𝑥 ~Q 𝑦)) |
139 | | fveq2 6768 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = 𝑦 → (2nd ‘𝑧) = (2nd ‘𝑦)) |
140 | 139 | breq1d 5085 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 𝑦 → ((2nd ‘𝑧) <N
(2nd ‘𝑥)
↔ (2nd ‘𝑦) <N
(2nd ‘𝑥))) |
141 | 140 | notbid 318 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝑦 → (¬ (2nd ‘𝑧) <N
(2nd ‘𝑥)
↔ ¬ (2nd ‘𝑦) <N
(2nd ‘𝑥))) |
142 | 138, 141 | imbi12d 345 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = 𝑦 → ((𝑥 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N (2nd ‘𝑥)) ↔ (𝑥 ~Q 𝑦 → ¬ (2nd
‘𝑦)
<N (2nd ‘𝑥)))) |
143 | 142 | rspcva 3559 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ (N ×
N) ∧ ∀𝑧 ∈ (N ×
N)(𝑥
~Q 𝑧 → ¬ (2nd ‘𝑧) <N
(2nd ‘𝑥)))
→ (𝑥
~Q 𝑦 → ¬ (2nd ‘𝑦) <N
(2nd ‘𝑥))) |
144 | 129, 137,
143 | syl2anr 597 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (𝑥
~Q 𝑦 → ¬ (2nd ‘𝑦) <N
(2nd ‘𝑥))) |
145 | 144 | imp 407 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → ¬ (2nd ‘𝑦) <N
(2nd ‘𝑥)) |
146 | | elpqn 10670 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ Q →
𝑥 ∈ (N
× N)) |
147 | 91 | rabeq2i 3421 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ Q ↔
(𝑦 ∈ (N
× N) ∧ ∀𝑧 ∈ (N ×
N)(𝑦
~Q 𝑧 → ¬ (2nd ‘𝑧) <N
(2nd ‘𝑦)))) |
148 | 147 | simprbi 497 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ Q →
∀𝑧 ∈
(N × N)(𝑦 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N (2nd ‘𝑦))) |
149 | | breq2 5079 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = 𝑥 → (𝑦 ~Q 𝑧 ↔ 𝑦 ~Q 𝑥)) |
150 | | fveq2 6768 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = 𝑥 → (2nd ‘𝑧) = (2nd ‘𝑥)) |
151 | 150 | breq1d 5085 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = 𝑥 → ((2nd ‘𝑧) <N
(2nd ‘𝑦)
↔ (2nd ‘𝑥) <N
(2nd ‘𝑦))) |
152 | 151 | notbid 318 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = 𝑥 → (¬ (2nd ‘𝑧) <N
(2nd ‘𝑦)
↔ ¬ (2nd ‘𝑥) <N
(2nd ‘𝑦))) |
153 | 149, 152 | imbi12d 345 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 𝑥 → ((𝑦 ~Q 𝑧 → ¬ (2nd
‘𝑧)
<N (2nd ‘𝑦)) ↔ (𝑦 ~Q 𝑥 → ¬ (2nd
‘𝑥)
<N (2nd ‘𝑦)))) |
154 | 153 | rspcva 3559 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ (N ×
N) ∧ ∀𝑧 ∈ (N ×
N)(𝑦
~Q 𝑧 → ¬ (2nd ‘𝑧) <N
(2nd ‘𝑦)))
→ (𝑦
~Q 𝑥 → ¬ (2nd ‘𝑥) <N
(2nd ‘𝑦))) |
155 | 146, 148,
154 | syl2an 596 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (𝑦
~Q 𝑥 → ¬ (2nd ‘𝑥) <N
(2nd ‘𝑦))) |
156 | 64 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ~Q
𝑦 →
~Q Er (N ×
N)) |
157 | | id 22 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ~Q
𝑦 → 𝑥 ~Q 𝑦) |
158 | 156, 157 | ersym 8499 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ~Q
𝑦 → 𝑦 ~Q 𝑥) |
159 | 155, 158 | impel 506 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → ¬ (2nd ‘𝑥) <N
(2nd ‘𝑦)) |
160 | | xp2nd 7855 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ (N ×
N) → (2nd ‘𝑥) ∈ N) |
161 | 146, 160 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ Q →
(2nd ‘𝑥)
∈ N) |
162 | 161 | ad2antrr 723 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → (2nd ‘𝑥) ∈
N) |
163 | | xp2nd 7855 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (N ×
N) → (2nd ‘𝑦) ∈ N) |
164 | 129, 163 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ Q →
(2nd ‘𝑦)
∈ N) |
165 | 164 | ad2antlr 724 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → (2nd ‘𝑦) ∈
N) |
166 | | ltsopi 10633 |
. . . . . . . . . . . . . . . . . . 19
⊢
<N Or N |
167 | | sotric 5528 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((
<N Or N ∧ ((2nd
‘𝑥) ∈
N ∧ (2nd ‘𝑦) ∈ N)) →
((2nd ‘𝑥)
<N (2nd ‘𝑦) ↔ ¬ ((2nd ‘𝑥) = (2nd ‘𝑦) ∨ (2nd
‘𝑦)
<N (2nd ‘𝑥)))) |
168 | 166, 167 | mpan 687 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘𝑥) ∈ N ∧
(2nd ‘𝑦)
∈ N) → ((2nd ‘𝑥) <N
(2nd ‘𝑦)
↔ ¬ ((2nd ‘𝑥) = (2nd ‘𝑦) ∨ (2nd ‘𝑦) <N
(2nd ‘𝑥)))) |
169 | 168 | notbid 318 |
. . . . . . . . . . . . . . . . 17
⊢
(((2nd ‘𝑥) ∈ N ∧
(2nd ‘𝑦)
∈ N) → (¬ (2nd ‘𝑥) <N
(2nd ‘𝑦)
↔ ¬ ¬ ((2nd ‘𝑥) = (2nd ‘𝑦) ∨ (2nd ‘𝑦) <N
(2nd ‘𝑥)))) |
170 | | notnotb 315 |
. . . . . . . . . . . . . . . . 17
⊢
(((2nd ‘𝑥) = (2nd ‘𝑦) ∨ (2nd ‘𝑦) <N
(2nd ‘𝑥))
↔ ¬ ¬ ((2nd ‘𝑥) = (2nd ‘𝑦) ∨ (2nd ‘𝑦) <N
(2nd ‘𝑥))) |
171 | 169, 170 | bitr4di 289 |
. . . . . . . . . . . . . . . 16
⊢
(((2nd ‘𝑥) ∈ N ∧
(2nd ‘𝑦)
∈ N) → (¬ (2nd ‘𝑥) <N
(2nd ‘𝑦)
↔ ((2nd ‘𝑥) = (2nd ‘𝑦) ∨ (2nd ‘𝑦) <N
(2nd ‘𝑥)))) |
172 | 162, 165,
171 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → (¬ (2nd ‘𝑥) <N
(2nd ‘𝑦)
↔ ((2nd ‘𝑥) = (2nd ‘𝑦) ∨ (2nd ‘𝑦) <N
(2nd ‘𝑥)))) |
173 | 159, 172 | mpbid 231 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → ((2nd ‘𝑥) = (2nd ‘𝑦) ∨ (2nd
‘𝑦)
<N (2nd ‘𝑥))) |
174 | 173 | ord 861 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → (¬ (2nd ‘𝑥) = (2nd ‘𝑦) → (2nd
‘𝑦)
<N (2nd ‘𝑥))) |
175 | 145, 174 | mt3d 148 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → (2nd ‘𝑥) = (2nd ‘𝑦)) |
176 | 175 | oveq2d 7285 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → ((1st ‘𝑥)
·N (2nd ‘𝑥)) = ((1st ‘𝑥)
·N (2nd ‘𝑦))) |
177 | 128, 176 | eqtrid 2790 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → ((2nd ‘𝑥)
·N (1st ‘𝑥)) = ((1st ‘𝑥)
·N (2nd ‘𝑦))) |
178 | | 1st2nd2 7861 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (N ×
N) → 𝑥 =
〈(1st ‘𝑥), (2nd ‘𝑥)〉) |
179 | | 1st2nd2 7861 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (N ×
N) → 𝑦 =
〈(1st ‘𝑦), (2nd ‘𝑦)〉) |
180 | 178, 179 | breqan12d 5091 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ (N ×
N) ∧ 𝑦
∈ (N × N)) → (𝑥 ~Q 𝑦 ↔ 〈(1st
‘𝑥), (2nd
‘𝑥)〉
~Q 〈(1st ‘𝑦), (2nd ‘𝑦)〉)) |
181 | | xp1st 7854 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (N ×
N) → (1st ‘𝑥) ∈ N) |
182 | 181, 160 | jca 512 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (N ×
N) → ((1st ‘𝑥) ∈ N ∧
(2nd ‘𝑥)
∈ N)) |
183 | | xp1st 7854 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (N ×
N) → (1st ‘𝑦) ∈ N) |
184 | 183, 163 | jca 512 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (N ×
N) → ((1st ‘𝑦) ∈ N ∧
(2nd ‘𝑦)
∈ N)) |
185 | | enqbreq 10664 |
. . . . . . . . . . . . . 14
⊢
((((1st ‘𝑥) ∈ N ∧
(2nd ‘𝑥)
∈ N) ∧ ((1st ‘𝑦) ∈ N ∧
(2nd ‘𝑦)
∈ N)) → (〈(1st ‘𝑥), (2nd ‘𝑥)〉
~Q 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ↔ ((1st ‘𝑥)
·N (2nd ‘𝑦)) = ((2nd ‘𝑥)
·N (1st ‘𝑦)))) |
186 | 182, 184,
185 | syl2an 596 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ (N ×
N) ∧ 𝑦
∈ (N × N)) → (〈(1st
‘𝑥), (2nd
‘𝑥)〉
~Q 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ↔ ((1st ‘𝑥)
·N (2nd ‘𝑦)) = ((2nd ‘𝑥)
·N (1st ‘𝑦)))) |
187 | 180, 186 | bitrd 278 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ (N ×
N) ∧ 𝑦
∈ (N × N)) → (𝑥 ~Q 𝑦 ↔ ((1st
‘𝑥)
·N (2nd ‘𝑦)) = ((2nd ‘𝑥)
·N (1st ‘𝑦)))) |
188 | 146, 129,
187 | syl2an 596 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (𝑥
~Q 𝑦 ↔ ((1st ‘𝑥)
·N (2nd ‘𝑦)) = ((2nd ‘𝑥)
·N (1st ‘𝑦)))) |
189 | 188 | biimpa 477 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → ((1st ‘𝑥)
·N (2nd ‘𝑦)) = ((2nd ‘𝑥)
·N (1st ‘𝑦))) |
190 | 177, 189 | eqtrd 2778 |
. . . . . . . . 9
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → ((2nd ‘𝑥)
·N (1st ‘𝑥)) = ((2nd ‘𝑥)
·N (1st ‘𝑦))) |
191 | 146 | ad2antrr 723 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → 𝑥 ∈ (N ×
N)) |
192 | | mulcanpi 10645 |
. . . . . . . . . . 11
⊢
(((2nd ‘𝑥) ∈ N ∧
(1st ‘𝑥)
∈ N) → (((2nd ‘𝑥) ·N
(1st ‘𝑥))
= ((2nd ‘𝑥) ·N
(1st ‘𝑦))
↔ (1st ‘𝑥) = (1st ‘𝑦))) |
193 | 160, 181,
192 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (N ×
N) → (((2nd ‘𝑥) ·N
(1st ‘𝑥))
= ((2nd ‘𝑥) ·N
(1st ‘𝑦))
↔ (1st ‘𝑥) = (1st ‘𝑦))) |
194 | 191, 193 | syl 17 |
. . . . . . . . 9
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → (((2nd ‘𝑥)
·N (1st ‘𝑥)) = ((2nd ‘𝑥)
·N (1st ‘𝑦)) ↔ (1st ‘𝑥) = (1st ‘𝑦))) |
195 | 190, 194 | mpbid 231 |
. . . . . . . 8
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → (1st ‘𝑥) = (1st ‘𝑦)) |
196 | 195, 175 | opeq12d 4814 |
. . . . . . 7
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → 〈(1st ‘𝑥), (2nd ‘𝑥)〉 = 〈(1st
‘𝑦), (2nd
‘𝑦)〉) |
197 | 191, 178 | syl 17 |
. . . . . . 7
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → 𝑥 = 〈(1st ‘𝑥), (2nd ‘𝑥)〉) |
198 | 129 | ad2antlr 724 |
. . . . . . . 8
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → 𝑦 ∈ (N ×
N)) |
199 | 198, 179 | syl 17 |
. . . . . . 7
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) |
200 | 196, 197,
199 | 3eqtr4d 2788 |
. . . . . 6
⊢ (((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ 𝑥
~Q 𝑦) → 𝑥 = 𝑦) |
201 | 200 | ex 413 |
. . . . 5
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (𝑥
~Q 𝑦 → 𝑥 = 𝑦)) |
202 | 127, 201 | syl5 34 |
. . . 4
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ ((𝑥
~Q 𝑎 ∧ 𝑦 ~Q 𝑎) → 𝑥 = 𝑦)) |
203 | 202 | rgen2 3128 |
. . 3
⊢
∀𝑥 ∈
Q ∀𝑦
∈ Q ((𝑥
~Q 𝑎 ∧ 𝑦 ~Q 𝑎) → 𝑥 = 𝑦) |
204 | 123, 203 | vtoclg 3504 |
. 2
⊢ (𝐴 ∈ (N ×
N) → ∀𝑥 ∈ Q ∀𝑦 ∈ Q ((𝑥 ~Q
𝐴 ∧ 𝑦 ~Q 𝐴) → 𝑥 = 𝑦)) |
205 | | breq1 5078 |
. . 3
⊢ (𝑥 = 𝑦 → (𝑥 ~Q 𝐴 ↔ 𝑦 ~Q 𝐴)) |
206 | 205 | reu4 3667 |
. 2
⊢
(∃!𝑥 ∈
Q 𝑥
~Q 𝐴 ↔ (∃𝑥 ∈ Q 𝑥 ~Q 𝐴 ∧ ∀𝑥 ∈ Q
∀𝑦 ∈
Q ((𝑥
~Q 𝐴 ∧ 𝑦 ~Q 𝐴) → 𝑥 = 𝑦))) |
207 | 118, 204,
206 | sylanbrc 583 |
1
⊢ (𝐴 ∈ (N ×
N) → ∃!𝑥 ∈ Q 𝑥 ~Q 𝐴) |