Step | Hyp | Ref
| Expression |
1 | | cltxr 36117 |
. 2
class
<ℝ̅ |
2 | | vx |
. . . . . . . . . . 11
setvar 𝑥 |
3 | 2 | cv 1540 |
. . . . . . . . . 10
class 𝑥 |
4 | | c1st 7972 |
. . . . . . . . . 10
class
1st |
5 | 3, 4 | cfv 6543 |
. . . . . . . . 9
class
(1st ‘𝑥) |
6 | | vy |
. . . . . . . . . . 11
setvar 𝑦 |
7 | 6 | cv 1540 |
. . . . . . . . . 10
class 𝑦 |
8 | | c0r 10860 |
. . . . . . . . . 10
class
0R |
9 | 7, 8 | cop 4634 |
. . . . . . . . 9
class
⟨𝑦,
0R⟩ |
10 | 5, 9 | wceq 1541 |
. . . . . . . 8
wff
(1st ‘𝑥) = ⟨𝑦,
0R⟩ |
11 | | c2nd 7973 |
. . . . . . . . . 10
class
2nd |
12 | 3, 11 | cfv 6543 |
. . . . . . . . 9
class
(2nd ‘𝑥) |
13 | | vz |
. . . . . . . . . . 11
setvar 𝑧 |
14 | 13 | cv 1540 |
. . . . . . . . . 10
class 𝑧 |
15 | 14, 8 | cop 4634 |
. . . . . . . . 9
class
⟨𝑧,
0R⟩ |
16 | 12, 15 | wceq 1541 |
. . . . . . . 8
wff
(2nd ‘𝑥) = ⟨𝑧,
0R⟩ |
17 | 10, 16 | wa 396 |
. . . . . . 7
wff
((1st ‘𝑥) = ⟨𝑦, 0R⟩ ∧
(2nd ‘𝑥) =
⟨𝑧,
0R⟩) |
18 | | cltr 10865 |
. . . . . . . 8
class
<R |
19 | 7, 14, 18 | wbr 5148 |
. . . . . . 7
wff 𝑦 <R
𝑧 |
20 | 17, 19 | wa 396 |
. . . . . 6
wff
(((1st ‘𝑥) = ⟨𝑦, 0R⟩ ∧
(2nd ‘𝑥) =
⟨𝑧,
0R⟩) ∧ 𝑦 <R 𝑧) |
21 | 20, 13 | wex 1781 |
. . . . 5
wff
∃𝑧(((1st ‘𝑥) = ⟨𝑦, 0R⟩ ∧
(2nd ‘𝑥) =
⟨𝑧,
0R⟩) ∧ 𝑦 <R 𝑧) |
22 | 21, 6 | wex 1781 |
. . . 4
wff
∃𝑦∃𝑧(((1st ‘𝑥) = ⟨𝑦, 0R⟩ ∧
(2nd ‘𝑥) =
⟨𝑧,
0R⟩) ∧ 𝑦 <R 𝑧) |
23 | | crrbar 36104 |
. . . . 5
class
ℝ̅ |
24 | 23, 23 | cxp 5674 |
. . . 4
class
(ℝ̅ × ℝ̅) |
25 | 22, 2, 24 | crab 3432 |
. . 3
class {𝑥 ∈ (ℝ̅ ×
ℝ̅) ∣ ∃𝑦∃𝑧(((1st ‘𝑥) = ⟨𝑦, 0R⟩ ∧
(2nd ‘𝑥) =
⟨𝑧,
0R⟩) ∧ 𝑦 <R 𝑧)} |
26 | | cminfty 36099 |
. . . . . . 7
class
-∞ |
27 | 26 | csn 4628 |
. . . . . 6
class
{-∞} |
28 | | cr 11108 |
. . . . . 6
class
ℝ |
29 | 27, 28 | cxp 5674 |
. . . . 5
class
({-∞} × ℝ) |
30 | | cpinfty 36095 |
. . . . . . 7
class
+∞ |
31 | 30 | csn 4628 |
. . . . . 6
class
{+∞} |
32 | 28, 31 | cxp 5674 |
. . . . 5
class (ℝ
× {+∞}) |
33 | 29, 32 | cun 3946 |
. . . 4
class
(({-∞} × ℝ) ∪ (ℝ ×
{+∞})) |
34 | 27, 31 | cxp 5674 |
. . . 4
class
({-∞} × {+∞}) |
35 | 33, 34 | cun 3946 |
. . 3
class
((({-∞} × ℝ) ∪ (ℝ × {+∞})) ∪
({-∞} × {+∞})) |
36 | 25, 35 | cun 3946 |
. 2
class ({𝑥 ∈ (ℝ̅ ×
ℝ̅) ∣ ∃𝑦∃𝑧(((1st ‘𝑥) = ⟨𝑦, 0R⟩ ∧
(2nd ‘𝑥) =
⟨𝑧,
0R⟩) ∧ 𝑦 <R 𝑧)} ∪ ((({-∞} ×
ℝ) ∪ (ℝ × {+∞})) ∪ ({-∞} ×
{+∞}))) |
37 | 1, 36 | wceq 1541 |
1
wff
<ℝ̅ = ({𝑥 ∈ (ℝ̅ ×
ℝ̅) ∣ ∃𝑦∃𝑧(((1st ‘𝑥) = ⟨𝑦, 0R⟩ ∧
(2nd ‘𝑥) =
⟨𝑧,
0R⟩) ∧ 𝑦 <R 𝑧)} ∪ ((({-∞} ×
ℝ) ∪ (ℝ × {+∞})) ∪ ({-∞} ×
{+∞}))) |