Step | Hyp | Ref
| Expression |
1 | | cfractemp 36077 |
. 2
class
{R |
2 | | vx |
. . 3
setvar 𝑥 |
3 | | cnr 10860 |
. . 3
class
R |
4 | | vy |
. . . . . . . 8
setvar 𝑦 |
5 | 4 | cv 1541 |
. . . . . . 7
class 𝑦 |
6 | | c0r 10861 |
. . . . . . 7
class
0R |
7 | 5, 6 | wceq 1542 |
. . . . . 6
wff 𝑦 =
0R |
8 | | cltr 10866 |
. . . . . . . 8
class
<R |
9 | 6, 5, 8 | wbr 5149 |
. . . . . . 7
wff
0R <R 𝑦 |
10 | | c1r 10862 |
. . . . . . . 8
class
1R |
11 | 5, 10, 8 | wbr 5149 |
. . . . . . 7
wff 𝑦 <R
1R |
12 | 9, 11 | wa 397 |
. . . . . 6
wff
(0R <R 𝑦 ∧ 𝑦 <R
1R) |
13 | 7, 12 | wo 846 |
. . . . 5
wff (𝑦 = 0R
∨ (0R <R 𝑦 ∧ 𝑦 <R
1R)) |
14 | | vz |
. . . . . . . . . . . . 13
setvar 𝑧 |
15 | 14 | cv 1541 |
. . . . . . . . . . . 12
class 𝑧 |
16 | | vn |
. . . . . . . . . . . . . . 15
setvar 𝑛 |
17 | 16 | cv 1541 |
. . . . . . . . . . . . . 14
class 𝑛 |
18 | 17 | csuc 6367 |
. . . . . . . . . . . . 13
class suc 𝑛 |
19 | | c1o 8459 |
. . . . . . . . . . . . 13
class
1o |
20 | 18, 19 | cop 4635 |
. . . . . . . . . . . 12
class ⟨suc
𝑛,
1o⟩ |
21 | | cltq 10853 |
. . . . . . . . . . . 12
class
<Q |
22 | 15, 20, 21 | wbr 5149 |
. . . . . . . . . . 11
wff 𝑧 <Q
⟨suc 𝑛,
1o⟩ |
23 | | cnq 10847 |
. . . . . . . . . . 11
class
Q |
24 | 22, 14, 23 | crab 3433 |
. . . . . . . . . 10
class {𝑧 ∈ Q ∣
𝑧
<Q ⟨suc 𝑛, 1o⟩} |
25 | | c1p 10855 |
. . . . . . . . . 10
class
1P |
26 | 24, 25 | cop 4635 |
. . . . . . . . 9
class
⟨{𝑧 ∈
Q ∣ 𝑧
<Q ⟨suc 𝑛, 1o⟩},
1P⟩ |
27 | | cer 10859 |
. . . . . . . . 9
class
~R |
28 | 26, 27 | cec 8701 |
. . . . . . . 8
class
[⟨{𝑧 ∈
Q ∣ 𝑧
<Q ⟨suc 𝑛, 1o⟩},
1P⟩] ~R |
29 | | cplr 10864 |
. . . . . . . 8
class
+R |
30 | 28, 5, 29 | co 7409 |
. . . . . . 7
class
([⟨{𝑧 ∈
Q ∣ 𝑧
<Q ⟨suc 𝑛, 1o⟩},
1P⟩] ~R
+R 𝑦) |
31 | 2 | cv 1541 |
. . . . . . 7
class 𝑥 |
32 | 30, 31 | wceq 1542 |
. . . . . 6
wff
([⟨{𝑧 ∈
Q ∣ 𝑧
<Q ⟨suc 𝑛, 1o⟩},
1P⟩] ~R
+R 𝑦) = 𝑥 |
33 | | com 7855 |
. . . . . 6
class
ω |
34 | 32, 16, 33 | wrex 3071 |
. . . . 5
wff
∃𝑛 ∈
ω ([⟨{𝑧 ∈
Q ∣ 𝑧
<Q ⟨suc 𝑛, 1o⟩},
1P⟩] ~R
+R 𝑦) = 𝑥 |
35 | 13, 34 | wa 397 |
. . . 4
wff ((𝑦 = 0R
∨ (0R <R 𝑦 ∧ 𝑦 <R
1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q
⟨suc 𝑛,
1o⟩}, 1P⟩]
~R +R 𝑦) = 𝑥) |
36 | 35, 4, 3 | crio 7364 |
. . 3
class
(℩𝑦
∈ R ((𝑦
= 0R ∨ (0R
<R 𝑦 ∧ 𝑦 <R
1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q
⟨suc 𝑛,
1o⟩}, 1P⟩]
~R +R 𝑦) = 𝑥)) |
37 | 2, 3, 36 | cmpt 5232 |
. 2
class (𝑥 ∈ R ↦
(℩𝑦 ∈
R ((𝑦 =
0R ∨ (0R
<R 𝑦 ∧ 𝑦 <R
1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q
⟨suc 𝑛,
1o⟩}, 1P⟩]
~R +R 𝑦) = 𝑥))) |
38 | 1, 37 | wceq 1542 |
1
wff
{R = (𝑥 ∈ R ↦
(℩𝑦 ∈
R ((𝑦 =
0R ∨ (0R
<R 𝑦 ∧ 𝑦 <R
1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q
⟨suc 𝑛,
1o⟩}, 1P⟩]
~R +R 𝑦) = 𝑥))) |