Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
|- ( x = (/) -> ( ( (/) Sat (/) ) ` x ) = ( ( (/) Sat (/) ) ` (/) ) ) |
2 |
1
|
eleq2d |
|- ( x = (/) -> ( (/) e. ( ( (/) Sat (/) ) ` x ) <-> (/) e. ( ( (/) Sat (/) ) ` (/) ) ) ) |
3 |
2
|
notbid |
|- ( x = (/) -> ( -. (/) e. ( ( (/) Sat (/) ) ` x ) <-> -. (/) e. ( ( (/) Sat (/) ) ` (/) ) ) ) |
4 |
|
fveq2 |
|- ( x = y -> ( ( (/) Sat (/) ) ` x ) = ( ( (/) Sat (/) ) ` y ) ) |
5 |
4
|
eleq2d |
|- ( x = y -> ( (/) e. ( ( (/) Sat (/) ) ` x ) <-> (/) e. ( ( (/) Sat (/) ) ` y ) ) ) |
6 |
5
|
notbid |
|- ( x = y -> ( -. (/) e. ( ( (/) Sat (/) ) ` x ) <-> -. (/) e. ( ( (/) Sat (/) ) ` y ) ) ) |
7 |
|
fveq2 |
|- ( x = suc y -> ( ( (/) Sat (/) ) ` x ) = ( ( (/) Sat (/) ) ` suc y ) ) |
8 |
7
|
eleq2d |
|- ( x = suc y -> ( (/) e. ( ( (/) Sat (/) ) ` x ) <-> (/) e. ( ( (/) Sat (/) ) ` suc y ) ) ) |
9 |
8
|
notbid |
|- ( x = suc y -> ( -. (/) e. ( ( (/) Sat (/) ) ` x ) <-> -. (/) e. ( ( (/) Sat (/) ) ` suc y ) ) ) |
10 |
|
fveq2 |
|- ( x = N -> ( ( (/) Sat (/) ) ` x ) = ( ( (/) Sat (/) ) ` N ) ) |
11 |
10
|
eleq2d |
|- ( x = N -> ( (/) e. ( ( (/) Sat (/) ) ` x ) <-> (/) e. ( ( (/) Sat (/) ) ` N ) ) ) |
12 |
11
|
notbid |
|- ( x = N -> ( -. (/) e. ( ( (/) Sat (/) ) ` x ) <-> -. (/) e. ( ( (/) Sat (/) ) ` N ) ) ) |
13 |
|
0nelopab |
|- -. (/) e. { <. x , y >. | ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) } |
14 |
|
satf00 |
|- ( ( (/) Sat (/) ) ` (/) ) = { <. x , y >. | ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) } |
15 |
14
|
eleq2i |
|- ( (/) e. ( ( (/) Sat (/) ) ` (/) ) <-> (/) e. { <. x , y >. | ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) } ) |
16 |
13 15
|
mtbir |
|- -. (/) e. ( ( (/) Sat (/) ) ` (/) ) |
17 |
|
simpr |
|- ( ( y e. _om /\ -. (/) e. ( ( (/) Sat (/) ) ` y ) ) -> -. (/) e. ( ( (/) Sat (/) ) ` y ) ) |
18 |
|
0nelopab |
|- -. (/) e. { <. x , z >. | ( z = (/) /\ E. u e. ( ( (/) Sat (/) ) ` y ) ( E. v e. ( ( (/) Sat (/) ) ` y ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } |
19 |
|
ioran |
|- ( -. ( (/) e. ( ( (/) Sat (/) ) ` y ) \/ (/) e. { <. x , z >. | ( z = (/) /\ E. u e. ( ( (/) Sat (/) ) ` y ) ( E. v e. ( ( (/) Sat (/) ) ` y ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) <-> ( -. (/) e. ( ( (/) Sat (/) ) ` y ) /\ -. (/) e. { <. x , z >. | ( z = (/) /\ E. u e. ( ( (/) Sat (/) ) ` y ) ( E. v e. ( ( (/) Sat (/) ) ` y ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) ) |
20 |
17 18 19
|
sylanblrc |
|- ( ( y e. _om /\ -. (/) e. ( ( (/) Sat (/) ) ` y ) ) -> -. ( (/) e. ( ( (/) Sat (/) ) ` y ) \/ (/) e. { <. x , z >. | ( z = (/) /\ E. u e. ( ( (/) Sat (/) ) ` y ) ( E. v e. ( ( (/) Sat (/) ) ` y ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) ) |
21 |
|
eqid |
|- ( (/) Sat (/) ) = ( (/) Sat (/) ) |
22 |
21
|
satf0suc |
|- ( y e. _om -> ( ( (/) Sat (/) ) ` suc y ) = ( ( ( (/) Sat (/) ) ` y ) u. { <. x , z >. | ( z = (/) /\ E. u e. ( ( (/) Sat (/) ) ` y ) ( E. v e. ( ( (/) Sat (/) ) ` y ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) ) |
23 |
22
|
adantr |
|- ( ( y e. _om /\ -. (/) e. ( ( (/) Sat (/) ) ` y ) ) -> ( ( (/) Sat (/) ) ` suc y ) = ( ( ( (/) Sat (/) ) ` y ) u. { <. x , z >. | ( z = (/) /\ E. u e. ( ( (/) Sat (/) ) ` y ) ( E. v e. ( ( (/) Sat (/) ) ` y ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) ) |
24 |
23
|
eleq2d |
|- ( ( y e. _om /\ -. (/) e. ( ( (/) Sat (/) ) ` y ) ) -> ( (/) e. ( ( (/) Sat (/) ) ` suc y ) <-> (/) e. ( ( ( (/) Sat (/) ) ` y ) u. { <. x , z >. | ( z = (/) /\ E. u e. ( ( (/) Sat (/) ) ` y ) ( E. v e. ( ( (/) Sat (/) ) ` y ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) ) ) |
25 |
|
elun |
|- ( (/) e. ( ( ( (/) Sat (/) ) ` y ) u. { <. x , z >. | ( z = (/) /\ E. u e. ( ( (/) Sat (/) ) ` y ) ( E. v e. ( ( (/) Sat (/) ) ` y ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) <-> ( (/) e. ( ( (/) Sat (/) ) ` y ) \/ (/) e. { <. x , z >. | ( z = (/) /\ E. u e. ( ( (/) Sat (/) ) ` y ) ( E. v e. ( ( (/) Sat (/) ) ` y ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) ) |
26 |
24 25
|
bitrdi |
|- ( ( y e. _om /\ -. (/) e. ( ( (/) Sat (/) ) ` y ) ) -> ( (/) e. ( ( (/) Sat (/) ) ` suc y ) <-> ( (/) e. ( ( (/) Sat (/) ) ` y ) \/ (/) e. { <. x , z >. | ( z = (/) /\ E. u e. ( ( (/) Sat (/) ) ` y ) ( E. v e. ( ( (/) Sat (/) ) ` y ) x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) ) ) |
27 |
20 26
|
mtbird |
|- ( ( y e. _om /\ -. (/) e. ( ( (/) Sat (/) ) ` y ) ) -> -. (/) e. ( ( (/) Sat (/) ) ` suc y ) ) |
28 |
27
|
ex |
|- ( y e. _om -> ( -. (/) e. ( ( (/) Sat (/) ) ` y ) -> -. (/) e. ( ( (/) Sat (/) ) ` suc y ) ) ) |
29 |
3 6 9 12 16 28
|
finds |
|- ( N e. _om -> -. (/) e. ( ( (/) Sat (/) ) ` N ) ) |
30 |
|
df-nel |
|- ( (/) e/ ( ( (/) Sat (/) ) ` N ) <-> -. (/) e. ( ( (/) Sat (/) ) ` N ) ) |
31 |
29 30
|
sylibr |
|- ( N e. _om -> (/) e/ ( ( (/) Sat (/) ) ` N ) ) |