Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. (Contributed by NM, 26-May-1998) Avoid ax-pow . (Revised by BTernaryTau, 23-Sep-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | phplem1 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl | |
|
2 | peano2 | |
|
3 | enrefnn | |
|
4 | 2 3 | syl | |
5 | 4 | adantr | |
6 | simpr | |
|
7 | dif1en | |
|
8 | 1 5 6 7 | syl3anc | |
9 | nnfi | |
|
10 | ensymfib | |
|
11 | 1 9 10 | 3syl | |
12 | 8 11 | mpbird | |