Step |
Hyp |
Ref |
Expression |
1 |
|
breq1 |
|- ( x = <. A , 1o >. -> ( x ~Q y <-> <. A , 1o >. ~Q y ) ) |
2 |
|
fveq2 |
|- ( x = <. A , 1o >. -> ( 2nd ` x ) = ( 2nd ` <. A , 1o >. ) ) |
3 |
2
|
breq2d |
|- ( x = <. A , 1o >. -> ( ( 2nd ` y ) ( 2nd ` y ) . ) ) ) |
4 |
3
|
notbid |
|- ( x = <. A , 1o >. -> ( -. ( 2nd ` y ) -. ( 2nd ` y ) . ) ) ) |
5 |
1 4
|
imbi12d |
|- ( x = <. A , 1o >. -> ( ( x ~Q y -> -. ( 2nd ` y ) ( <. A , 1o >. ~Q y -> -. ( 2nd ` y ) . ) ) ) ) |
6 |
5
|
ralbidv |
|- ( x = <. A , 1o >. -> ( A. y e. ( N. X. N. ) ( x ~Q y -> -. ( 2nd ` y ) A. y e. ( N. X. N. ) ( <. A , 1o >. ~Q y -> -. ( 2nd ` y ) . ) ) ) ) |
7 |
|
1pi |
|- 1o e. N. |
8 |
|
opelxpi |
|- ( ( A e. N. /\ 1o e. N. ) -> <. A , 1o >. e. ( N. X. N. ) ) |
9 |
7 8
|
mpan2 |
|- ( A e. N. -> <. A , 1o >. e. ( N. X. N. ) ) |
10 |
|
nlt1pi |
|- -. ( 2nd ` y ) |
11 |
|
1oex |
|- 1o e. _V |
12 |
|
op2ndg |
|- ( ( A e. N. /\ 1o e. _V ) -> ( 2nd ` <. A , 1o >. ) = 1o ) |
13 |
11 12
|
mpan2 |
|- ( A e. N. -> ( 2nd ` <. A , 1o >. ) = 1o ) |
14 |
13
|
breq2d |
|- ( A e. N. -> ( ( 2nd ` y ) . ) <-> ( 2nd ` y ) |
15 |
10 14
|
mtbiri |
|- ( A e. N. -> -. ( 2nd ` y ) . ) ) |
16 |
15
|
a1d |
|- ( A e. N. -> ( <. A , 1o >. ~Q y -> -. ( 2nd ` y ) . ) ) ) |
17 |
16
|
ralrimivw |
|- ( A e. N. -> A. y e. ( N. X. N. ) ( <. A , 1o >. ~Q y -> -. ( 2nd ` y ) . ) ) ) |
18 |
6 9 17
|
elrabd |
|- ( A e. N. -> <. A , 1o >. e. { x e. ( N. X. N. ) | A. y e. ( N. X. N. ) ( x ~Q y -> -. ( 2nd ` y ) |
19 |
|
df-nq |
|- Q. = { x e. ( N. X. N. ) | A. y e. ( N. X. N. ) ( x ~Q y -> -. ( 2nd ` y ) |
20 |
18 19
|
eleqtrrdi |
|- ( A e. N. -> <. A , 1o >. e. Q. ) |