![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ax-pre-suploc | Unicode version |
Description: An inhabited,
bounded-above, located set of reals has a supremum.
Locatedness here means that given Although this and ax-caucvg 7944 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 7944. (Contributed by Jim Kingdon, 23-Jan-2024.) |
Ref | Expression |
---|---|
ax-pre-suploc |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA |
. . . . 5
![]() ![]() | |
2 | cr 7823 |
. . . . 5
![]() ![]() | |
3 | 1, 2 | wss 3141 |
. . . 4
![]() ![]() ![]() ![]() |
4 | vx |
. . . . . . 7
![]() ![]() | |
5 | 4 | cv 1362 |
. . . . . 6
![]() ![]() |
6 | 5, 1 | wcel 2158 |
. . . . 5
![]() ![]() ![]() ![]() |
7 | 6, 4 | wex 1502 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() |
8 | 3, 7 | wa 104 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | vy |
. . . . . . . 8
![]() ![]() | |
10 | 9 | cv 1362 |
. . . . . . 7
![]() ![]() |
11 | cltrr 7828 |
. . . . . . 7
![]() ![]() | |
12 | 10, 5, 11 | wbr 4015 |
. . . . . 6
![]() ![]() ![]() ![]() |
13 | 12, 9, 1 | wral 2465 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 13, 4, 2 | wrex 2466 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 5, 10, 11 | wbr 4015 |
. . . . . . 7
![]() ![]() ![]() ![]() |
16 | vz |
. . . . . . . . . . 11
![]() ![]() | |
17 | 16 | cv 1362 |
. . . . . . . . . 10
![]() ![]() |
18 | 5, 17, 11 | wbr 4015 |
. . . . . . . . 9
![]() ![]() ![]() ![]() |
19 | 18, 16, 1 | wrex 2466 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 17, 10, 11 | wbr 4015 |
. . . . . . . . 9
![]() ![]() ![]() ![]() |
21 | 20, 16, 1 | wral 2465 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 19, 21 | wo 709 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 15, 22 | wi 4 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23, 9, 2 | wral 2465 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24, 4, 2 | wral 2465 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 14, 25 | wa 104 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 8, 26 | wa 104 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 15 | wn 3 |
. . . . 5
![]() ![]() ![]() ![]() ![]() |
29 | 28, 9, 1 | wral 2465 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 10, 17, 11 | wbr 4015 |
. . . . . . 7
![]() ![]() ![]() ![]() |
31 | 30, 16, 1 | wrex 2466 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 12, 31 | wi 4 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 32, 9, 2 | wral 2465 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 29, 33 | wa 104 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 34, 4, 2 | wrex 2466 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 27, 35 | wi 4 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
This axiom is referenced by: axsuploc 8043 |
Copyright terms: Public domain | W3C validator |