Description: There are at least aleph-one irrationals. (Contributed by NM, 2-Feb-2005)
Ref | Expression | ||
---|---|---|---|
Assertion | aleph1irr | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aleph1re | |
|
2 | reex | |
|
3 | numth3 | |
|
4 | 2 3 | ax-mp | |
5 | nnenom | |
|
6 | 5 | ensymi | |
7 | ruc | |
|
8 | ensdomtr | |
|
9 | 6 7 8 | mp2an | |
10 | sdomdom | |
|
11 | 9 10 | ax-mp | |
12 | resdomq | |
|
13 | infdif | |
|
14 | 4 11 12 13 | mp3an | |
15 | 14 | ensymi | |
16 | domentr | |
|
17 | 1 15 16 | mp2an | |