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 , either there is an element of the set greater than , or is an upper bound. Although this and ax-caucvg 7887 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 7887. (Contributed by Jim Kingdon, 23-Jan-2024.) |
Ref | Expression |
---|---|
ax-pre-suploc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . . 5 | |
2 | cr 7766 | . . . . 5 | |
3 | 1, 2 | wss 3121 | . . . 4 |
4 | vx | . . . . . . 7 | |
5 | 4 | cv 1347 | . . . . . 6 |
6 | 5, 1 | wcel 2141 | . . . . 5 |
7 | 6, 4 | wex 1485 | . . . 4 |
8 | 3, 7 | wa 103 | . . 3 |
9 | vy | . . . . . . . 8 | |
10 | 9 | cv 1347 | . . . . . . 7 |
11 | cltrr 7771 | . . . . . . 7 | |
12 | 10, 5, 11 | wbr 3987 | . . . . . 6 |
13 | 12, 9, 1 | wral 2448 | . . . . 5 |
14 | 13, 4, 2 | wrex 2449 | . . . 4 |
15 | 5, 10, 11 | wbr 3987 | . . . . . . 7 |
16 | vz | . . . . . . . . . . 11 | |
17 | 16 | cv 1347 | . . . . . . . . . 10 |
18 | 5, 17, 11 | wbr 3987 | . . . . . . . . 9 |
19 | 18, 16, 1 | wrex 2449 | . . . . . . . 8 |
20 | 17, 10, 11 | wbr 3987 | . . . . . . . . 9 |
21 | 20, 16, 1 | wral 2448 | . . . . . . . 8 |
22 | 19, 21 | wo 703 | . . . . . . 7 |
23 | 15, 22 | wi 4 | . . . . . 6 |
24 | 23, 9, 2 | wral 2448 | . . . . 5 |
25 | 24, 4, 2 | wral 2448 | . . . 4 |
26 | 14, 25 | wa 103 | . . 3 |
27 | 8, 26 | wa 103 | . 2 |
28 | 15 | wn 3 | . . . . 5 |
29 | 28, 9, 1 | wral 2448 | . . . 4 |
30 | 10, 17, 11 | wbr 3987 | . . . . . . 7 |
31 | 30, 16, 1 | wrex 2449 | . . . . . 6 |
32 | 12, 31 | wi 4 | . . . . 5 |
33 | 32, 9, 2 | wral 2448 | . . . 4 |
34 | 29, 33 | wa 103 | . . 3 |
35 | 34, 4, 2 | wrex 2449 | . 2 |
36 | 27, 35 | wi 4 | 1 |
Colors of variables: wff set class |
This axiom is referenced by: axsuploc 7985 |
Copyright terms: Public domain | W3C validator |