Description: 0lt1 without ax-mulcom . (Contributed by SN, 13-Feb-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | sn-0lt1 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1ne0 | |
|
2 | 1re | |
|
3 | 0re | |
|
4 | 2 3 | lttri2i | |
5 | 1 4 | mpbi | |
6 | rernegcl | |
|
7 | 2 6 | mp1i | |
8 | relt0neg1 | |
|
9 | 2 8 | ax-mp | |
10 | 9 | biimpi | |
11 | 7 7 10 10 | mulgt0d | |
12 | 1red | |
|
13 | 6 12 | remulneg2d | |
14 | ax-1rid | |
|
15 | 6 14 | syl | |
16 | 15 | oveq2d | |
17 | renegneg | |
|
18 | 13 16 17 | 3eqtrd | |
19 | 2 18 | ax-mp | |
20 | 11 19 | breqtrdi | |
21 | id | |
|
22 | 20 21 | jaoi | |
23 | 5 22 | ax-mp | |