Step |
Hyp |
Ref |
Expression |
1 |
|
eldifsn |
|- ( N e. ( Prime \ { 2 } ) <-> ( N e. Prime /\ N =/= 2 ) ) |
2 |
|
prmz |
|- ( N e. Prime -> N e. ZZ ) |
3 |
2
|
adantr |
|- ( ( N e. Prime /\ N =/= 2 ) -> N e. ZZ ) |
4 |
|
necom |
|- ( N =/= 2 <-> 2 =/= N ) |
5 |
|
df-ne |
|- ( 2 =/= N <-> -. 2 = N ) |
6 |
4 5
|
sylbb |
|- ( N =/= 2 -> -. 2 = N ) |
7 |
6
|
adantl |
|- ( ( N e. Prime /\ N =/= 2 ) -> -. 2 = N ) |
8 |
|
1ne2 |
|- 1 =/= 2 |
9 |
8
|
nesymi |
|- -. 2 = 1 |
10 |
9
|
a1i |
|- ( ( N e. Prime /\ N =/= 2 ) -> -. 2 = 1 ) |
11 |
|
ioran |
|- ( -. ( 2 = N \/ 2 = 1 ) <-> ( -. 2 = N /\ -. 2 = 1 ) ) |
12 |
7 10 11
|
sylanbrc |
|- ( ( N e. Prime /\ N =/= 2 ) -> -. ( 2 = N \/ 2 = 1 ) ) |
13 |
|
2nn |
|- 2 e. NN |
14 |
13
|
a1i |
|- ( N =/= 2 -> 2 e. NN ) |
15 |
|
dvdsprime |
|- ( ( N e. Prime /\ 2 e. NN ) -> ( 2 || N <-> ( 2 = N \/ 2 = 1 ) ) ) |
16 |
14 15
|
sylan2 |
|- ( ( N e. Prime /\ N =/= 2 ) -> ( 2 || N <-> ( 2 = N \/ 2 = 1 ) ) ) |
17 |
12 16
|
mtbird |
|- ( ( N e. Prime /\ N =/= 2 ) -> -. 2 || N ) |
18 |
|
isodd3 |
|- ( N e. Odd <-> ( N e. ZZ /\ -. 2 || N ) ) |
19 |
3 17 18
|
sylanbrc |
|- ( ( N e. Prime /\ N =/= 2 ) -> N e. Odd ) |
20 |
1 19
|
sylbi |
|- ( N e. ( Prime \ { 2 } ) -> N e. Odd ) |