Step |
Hyp |
Ref |
Expression |
1 |
|
olc |
|- ( N = 1 -> ( N = 0 \/ N = 1 ) ) |
2 |
1
|
a1d |
|- ( N = 1 -> ( ( N e. NN0 /\ N < 2 ) -> ( N = 0 \/ N = 1 ) ) ) |
3 |
|
nn0z |
|- ( N e. NN0 -> N e. ZZ ) |
4 |
|
2z |
|- 2 e. ZZ |
5 |
|
zltlem1 |
|- ( ( N e. ZZ /\ 2 e. ZZ ) -> ( N < 2 <-> N <_ ( 2 - 1 ) ) ) |
6 |
3 4 5
|
sylancl |
|- ( N e. NN0 -> ( N < 2 <-> N <_ ( 2 - 1 ) ) ) |
7 |
|
2m1e1 |
|- ( 2 - 1 ) = 1 |
8 |
7
|
breq2i |
|- ( N <_ ( 2 - 1 ) <-> N <_ 1 ) |
9 |
6 8
|
bitrdi |
|- ( N e. NN0 -> ( N < 2 <-> N <_ 1 ) ) |
10 |
|
necom |
|- ( N =/= 1 <-> 1 =/= N ) |
11 |
|
nn0re |
|- ( N e. NN0 -> N e. RR ) |
12 |
|
1re |
|- 1 e. RR |
13 |
|
ltlen |
|- ( ( N e. RR /\ 1 e. RR ) -> ( N < 1 <-> ( N <_ 1 /\ 1 =/= N ) ) ) |
14 |
11 12 13
|
sylancl |
|- ( N e. NN0 -> ( N < 1 <-> ( N <_ 1 /\ 1 =/= N ) ) ) |
15 |
|
nn0lt10b |
|- ( N e. NN0 -> ( N < 1 <-> N = 0 ) ) |
16 |
15
|
biimpa |
|- ( ( N e. NN0 /\ N < 1 ) -> N = 0 ) |
17 |
16
|
orcd |
|- ( ( N e. NN0 /\ N < 1 ) -> ( N = 0 \/ N = 1 ) ) |
18 |
17
|
ex |
|- ( N e. NN0 -> ( N < 1 -> ( N = 0 \/ N = 1 ) ) ) |
19 |
14 18
|
sylbird |
|- ( N e. NN0 -> ( ( N <_ 1 /\ 1 =/= N ) -> ( N = 0 \/ N = 1 ) ) ) |
20 |
19
|
expd |
|- ( N e. NN0 -> ( N <_ 1 -> ( 1 =/= N -> ( N = 0 \/ N = 1 ) ) ) ) |
21 |
10 20
|
syl7bi |
|- ( N e. NN0 -> ( N <_ 1 -> ( N =/= 1 -> ( N = 0 \/ N = 1 ) ) ) ) |
22 |
9 21
|
sylbid |
|- ( N e. NN0 -> ( N < 2 -> ( N =/= 1 -> ( N = 0 \/ N = 1 ) ) ) ) |
23 |
22
|
imp |
|- ( ( N e. NN0 /\ N < 2 ) -> ( N =/= 1 -> ( N = 0 \/ N = 1 ) ) ) |
24 |
23
|
com12 |
|- ( N =/= 1 -> ( ( N e. NN0 /\ N < 2 ) -> ( N = 0 \/ N = 1 ) ) ) |
25 |
2 24
|
pm2.61ine |
|- ( ( N e. NN0 /\ N < 2 ) -> ( N = 0 \/ N = 1 ) ) |