Step |
Hyp |
Ref |
Expression |
1 |
|
eldif |
|- ( A e. ( CC \ ( ZZ \ NN ) ) <-> ( A e. CC /\ -. A e. ( ZZ \ NN ) ) ) |
2 |
|
eldif |
|- ( A e. ( ZZ \ NN ) <-> ( A e. ZZ /\ -. A e. NN ) ) |
3 |
|
elznn |
|- ( A e. ZZ <-> ( A e. RR /\ ( A e. NN \/ -u A e. NN0 ) ) ) |
4 |
3
|
simprbi |
|- ( A e. ZZ -> ( A e. NN \/ -u A e. NN0 ) ) |
5 |
4
|
orcanai |
|- ( ( A e. ZZ /\ -. A e. NN ) -> -u A e. NN0 ) |
6 |
|
negneg |
|- ( A e. CC -> -u -u A = A ) |
7 |
6
|
adantr |
|- ( ( A e. CC /\ -u A e. NN0 ) -> -u -u A = A ) |
8 |
|
nn0negz |
|- ( -u A e. NN0 -> -u -u A e. ZZ ) |
9 |
8
|
adantl |
|- ( ( A e. CC /\ -u A e. NN0 ) -> -u -u A e. ZZ ) |
10 |
7 9
|
eqeltrrd |
|- ( ( A e. CC /\ -u A e. NN0 ) -> A e. ZZ ) |
11 |
10
|
ex |
|- ( A e. CC -> ( -u A e. NN0 -> A e. ZZ ) ) |
12 |
|
nngt0 |
|- ( A e. NN -> 0 < A ) |
13 |
|
nnre |
|- ( A e. NN -> A e. RR ) |
14 |
13
|
lt0neg2d |
|- ( A e. NN -> ( 0 < A <-> -u A < 0 ) ) |
15 |
12 14
|
mpbid |
|- ( A e. NN -> -u A < 0 ) |
16 |
13
|
renegcld |
|- ( A e. NN -> -u A e. RR ) |
17 |
|
0re |
|- 0 e. RR |
18 |
|
ltnle |
|- ( ( -u A e. RR /\ 0 e. RR ) -> ( -u A < 0 <-> -. 0 <_ -u A ) ) |
19 |
16 17 18
|
sylancl |
|- ( A e. NN -> ( -u A < 0 <-> -. 0 <_ -u A ) ) |
20 |
15 19
|
mpbid |
|- ( A e. NN -> -. 0 <_ -u A ) |
21 |
|
nn0ge0 |
|- ( -u A e. NN0 -> 0 <_ -u A ) |
22 |
20 21
|
nsyl3 |
|- ( -u A e. NN0 -> -. A e. NN ) |
23 |
11 22
|
jca2 |
|- ( A e. CC -> ( -u A e. NN0 -> ( A e. ZZ /\ -. A e. NN ) ) ) |
24 |
5 23
|
impbid2 |
|- ( A e. CC -> ( ( A e. ZZ /\ -. A e. NN ) <-> -u A e. NN0 ) ) |
25 |
2 24
|
syl5bb |
|- ( A e. CC -> ( A e. ( ZZ \ NN ) <-> -u A e. NN0 ) ) |
26 |
25
|
notbid |
|- ( A e. CC -> ( -. A e. ( ZZ \ NN ) <-> -. -u A e. NN0 ) ) |
27 |
26
|
pm5.32i |
|- ( ( A e. CC /\ -. A e. ( ZZ \ NN ) ) <-> ( A e. CC /\ -. -u A e. NN0 ) ) |
28 |
1 27
|
bitri |
|- ( A e. ( CC \ ( ZZ \ NN ) ) <-> ( A e. CC /\ -. -u A e. NN0 ) ) |