Description: Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012)
Ref | Expression | ||
---|---|---|---|
Assertion | eluz2b1 | |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. ZZ /\ 1 < N ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z | |- 2 e. ZZ |
|
2 | 1 | eluz1i | |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. ZZ /\ 2 <_ N ) ) |
3 | 1z | |- 1 e. ZZ |
|
4 | zltp1le | |- ( ( 1 e. ZZ /\ N e. ZZ ) -> ( 1 < N <-> ( 1 + 1 ) <_ N ) ) |
|
5 | 3 4 | mpan | |- ( N e. ZZ -> ( 1 < N <-> ( 1 + 1 ) <_ N ) ) |
6 | df-2 | |- 2 = ( 1 + 1 ) |
|
7 | 6 | breq1i | |- ( 2 <_ N <-> ( 1 + 1 ) <_ N ) |
8 | 5 7 | bitr4di | |- ( N e. ZZ -> ( 1 < N <-> 2 <_ N ) ) |
9 | 8 | pm5.32i | |- ( ( N e. ZZ /\ 1 < N ) <-> ( N e. ZZ /\ 2 <_ N ) ) |
10 | 2 9 | bitr4i | |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. ZZ /\ 1 < N ) ) |