Step |
Hyp |
Ref |
Expression |
1 |
|
retop |
|- ( topGen ` ran (,) ) e. Top |
2 |
|
uniretop |
|- RR = U. ( topGen ` ran (,) ) |
3 |
2
|
restid |
|- ( ( topGen ` ran (,) ) e. Top -> ( ( topGen ` ran (,) ) |`t RR ) = ( topGen ` ran (,) ) ) |
4 |
1 3
|
ax-mp |
|- ( ( topGen ` ran (,) ) |`t RR ) = ( topGen ` ran (,) ) |
5 |
|
iccssre |
|- ( ( x e. RR /\ y e. RR ) -> ( x [,] y ) C_ RR ) |
6 |
5
|
rgen2 |
|- A. x e. RR A. y e. RR ( x [,] y ) C_ RR |
7 |
|
ssid |
|- RR C_ RR |
8 |
|
reconn |
|- ( RR C_ RR -> ( ( ( topGen ` ran (,) ) |`t RR ) e. Conn <-> A. x e. RR A. y e. RR ( x [,] y ) C_ RR ) ) |
9 |
7 8
|
ax-mp |
|- ( ( ( topGen ` ran (,) ) |`t RR ) e. Conn <-> A. x e. RR A. y e. RR ( x [,] y ) C_ RR ) |
10 |
6 9
|
mpbir |
|- ( ( topGen ` ran (,) ) |`t RR ) e. Conn |
11 |
4 10
|
eqeltrri |
|- ( topGen ` ran (,) ) e. Conn |