Description: A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem using infima; this theorem removes the requirement that A be nonempty. (Contributed by Stefan O'Rear, 19-Oct-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | rencldnfi | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 | |
|
2 | simpl2 | |
|
3 | rexn0 | |
|
4 | 3 | ralimi | |
5 | 1rp | |
|
6 | ne0i | |
|
7 | r19.3rzv | |
|
8 | 5 6 7 | mp2b | |
9 | 4 8 | sylibr | |
10 | 9 | adantl | |
11 | simpl3 | |
|
12 | 10 11 | jca | |
13 | simpr | |
|
14 | rencldnfilem | |
|
15 | 1 2 12 13 14 | syl31anc | |