Step |
Hyp |
Ref |
Expression |
1 |
|
nnex |
|- NN e. _V |
2 |
1
|
canth2 |
|- NN ~< ~P NN |
3 |
|
domnsym |
|- ( ~P NN ~<_ NN -> -. NN ~< ~P NN ) |
4 |
2 3
|
mt2 |
|- -. ~P NN ~<_ NN |
5 |
|
retop |
|- ( topGen ` ran (,) ) e. Top |
6 |
|
simpl |
|- ( ( A C_ RR /\ A ~<_ NN ) -> A C_ RR ) |
7 |
|
uniretop |
|- RR = U. ( topGen ` ran (,) ) |
8 |
7
|
ntropn |
|- ( ( ( topGen ` ran (,) ) e. Top /\ A C_ RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) e. ( topGen ` ran (,) ) ) |
9 |
5 6 8
|
sylancr |
|- ( ( A C_ RR /\ A ~<_ NN ) -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) e. ( topGen ` ran (,) ) ) |
10 |
|
opnreen |
|- ( ( ( ( int ` ( topGen ` ran (,) ) ) ` A ) e. ( topGen ` ran (,) ) /\ ( ( int ` ( topGen ` ran (,) ) ) ` A ) =/= (/) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) ~~ ~P NN ) |
11 |
10
|
ex |
|- ( ( ( int ` ( topGen ` ran (,) ) ) ` A ) e. ( topGen ` ran (,) ) -> ( ( ( int ` ( topGen ` ran (,) ) ) ` A ) =/= (/) -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) ~~ ~P NN ) ) |
12 |
9 11
|
syl |
|- ( ( A C_ RR /\ A ~<_ NN ) -> ( ( ( int ` ( topGen ` ran (,) ) ) ` A ) =/= (/) -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) ~~ ~P NN ) ) |
13 |
|
reex |
|- RR e. _V |
14 |
13
|
ssex |
|- ( A C_ RR -> A e. _V ) |
15 |
7
|
ntrss2 |
|- ( ( ( topGen ` ran (,) ) e. Top /\ A C_ RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) C_ A ) |
16 |
5 15
|
mpan |
|- ( A C_ RR -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) C_ A ) |
17 |
|
ssdomg |
|- ( A e. _V -> ( ( ( int ` ( topGen ` ran (,) ) ) ` A ) C_ A -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) ~<_ A ) ) |
18 |
14 16 17
|
sylc |
|- ( A C_ RR -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) ~<_ A ) |
19 |
|
domtr |
|- ( ( ( ( int ` ( topGen ` ran (,) ) ) ` A ) ~<_ A /\ A ~<_ NN ) -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) ~<_ NN ) |
20 |
18 19
|
sylan |
|- ( ( A C_ RR /\ A ~<_ NN ) -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) ~<_ NN ) |
21 |
|
ensym |
|- ( ( ( int ` ( topGen ` ran (,) ) ) ` A ) ~~ ~P NN -> ~P NN ~~ ( ( int ` ( topGen ` ran (,) ) ) ` A ) ) |
22 |
|
endomtr |
|- ( ( ~P NN ~~ ( ( int ` ( topGen ` ran (,) ) ) ` A ) /\ ( ( int ` ( topGen ` ran (,) ) ) ` A ) ~<_ NN ) -> ~P NN ~<_ NN ) |
23 |
22
|
expcom |
|- ( ( ( int ` ( topGen ` ran (,) ) ) ` A ) ~<_ NN -> ( ~P NN ~~ ( ( int ` ( topGen ` ran (,) ) ) ` A ) -> ~P NN ~<_ NN ) ) |
24 |
20 21 23
|
syl2im |
|- ( ( A C_ RR /\ A ~<_ NN ) -> ( ( ( int ` ( topGen ` ran (,) ) ) ` A ) ~~ ~P NN -> ~P NN ~<_ NN ) ) |
25 |
12 24
|
syld |
|- ( ( A C_ RR /\ A ~<_ NN ) -> ( ( ( int ` ( topGen ` ran (,) ) ) ` A ) =/= (/) -> ~P NN ~<_ NN ) ) |
26 |
25
|
necon1bd |
|- ( ( A C_ RR /\ A ~<_ NN ) -> ( -. ~P NN ~<_ NN -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) = (/) ) ) |
27 |
4 26
|
mpi |
|- ( ( A C_ RR /\ A ~<_ NN ) -> ( ( int ` ( topGen ` ran (,) ) ) ` A ) = (/) ) |