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Theorem intvconlem1 25703
Description: All the intervals of  RR are connected. (Contributed by FL, 29-May-2014.)
Assertion
Ref Expression
intvconlem1  |-  ( I  e.  Intvl  ->  ( ( topGen `
 ran  (,) )t  I
)  e.  Con )

Proof of Theorem intvconlem1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3358 . . 3  |-  ( I  e.  ( ( ran 
(,)  u.  ( ran  (,] 
u.  ( ran  [,)  u. 
ran  [,] ) ) )  i^i  ~P RR )  <-> 
( I  e.  ( ran  (,)  u.  ( ran  (,]  u.  ( ran 
[,)  u.  ran  [,] )
) )  /\  I  e.  ~P RR ) )
2 elun 3316 . . . . 5  |-  ( I  e.  ( ran  (,)  u.  ( ran  (,]  u.  ( ran  [,)  u.  ran  [,] ) ) )  <->  ( I  e.  ran  (,)  \/  I  e.  ( ran  (,]  u.  ( ran  [,)  u.  ran  [,] ) ) ) )
3 bsi 25501 . . . . . . 7  |-  ( I  e.  ran  (,)  <->  E. x  e.  RR*  E. y  e. 
RR*  I  =  ( x (,) y ) )
4 icccon4 25702 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( topGen `  ran  (,) )t  (
x (,) y ) )  e.  Con )
54a1d 22 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( x (,) y
)  e.  ~P RR  ->  ( ( topGen `  ran  (,) )t  ( x (,) y
) )  e.  Con ) )
6 eleq1 2343 . . . . . . . . . 10  |-  ( I  =  ( x (,) y )  ->  (
I  e.  ~P RR  <->  ( x (,) y )  e.  ~P RR ) )
7 oveq2 5866 . . . . . . . . . . 11  |-  ( I  =  ( x (,) y )  ->  (
( topGen `  ran  (,) )t  I
)  =  ( (
topGen `  ran  (,) )t  (
x (,) y ) ) )
87eleq1d 2349 . . . . . . . . . 10  |-  ( I  =  ( x (,) y )  ->  (
( ( topGen `  ran  (,) )t  I )  e.  Con  <->  (
( topGen `  ran  (,) )t  (
x (,) y ) )  e.  Con )
)
96, 8imbi12d 311 . . . . . . . . 9  |-  ( I  =  ( x (,) y )  ->  (
( I  e.  ~P RR  ->  ( ( topGen ` 
ran  (,) )t  I )  e.  Con ) 
<->  ( ( x (,) y )  e.  ~P RR  ->  ( ( topGen ` 
ran  (,) )t  ( x (,) y ) )  e. 
Con ) ) )
105, 9syl5ibrcom 213 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
I  =  ( x (,) y )  -> 
( I  e.  ~P RR  ->  ( ( topGen ` 
ran  (,) )t  I )  e.  Con ) ) )
1110rexlimivv 2672 . . . . . . 7  |-  ( E. x  e.  RR*  E. y  e.  RR*  I  =  ( x (,) y )  ->  ( I  e. 
~P RR  ->  (
( topGen `  ran  (,) )t  I
)  e.  Con )
)
123, 11sylbi 187 . . . . . 6  |-  ( I  e.  ran  (,)  ->  ( I  e.  ~P RR  ->  ( ( topGen `  ran  (,) )t  I )  e.  Con ) )
13 elun 3316 . . . . . . 7  |-  ( I  e.  ( ran  (,]  u.  ( ran  [,)  u.  ran  [,] ) )  <->  ( I  e.  ran  (,]  \/  I  e.  ( ran  [,)  u.  ran  [,] ) ) )
14 bsi4 25643 . . . . . . . . 9  |-  ( I  e.  ran  (,]  <->  E. x  e.  RR*  E. y  e. 
RR*  I  =  ( x (,] y ) )
15 ovex 5883 . . . . . . . . . . . . . 14  |-  ( x (,] y )  e. 
_V
1615elpw 3631 . . . . . . . . . . . . 13  |-  ( ( x (,] y )  e.  ~P RR  <->  ( x (,] y )  C_  RR )
17 ubioc1 10705 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  < 
y )  ->  y  e.  ( x (,] y
) )
18 ssel2 3175 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x (,] y
)  C_  RR  /\  y  e.  ( x (,] y
) )  ->  y  e.  RR )
19 icccon3 25701 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  RR*  /\  y  e.  RR )  ->  (
( topGen `  ran  (,) )t  (
x (,] y ) )  e.  Con )
2019ex 423 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  e.  RR*  ->  ( y  e.  RR  ->  (
( topGen `  ran  (,) )t  (
x (,] y ) )  e.  Con )
)
2120adantr 451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  e.  RR  ->  ( ( topGen `  ran  (,) )t  (
x (,] y ) )  e.  Con )
)
2218, 21syl5com 26 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( x (,] y
)  C_  RR  /\  y  e.  ( x (,] y
) )  ->  (
( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( topGen `  ran  (,) )t  ( x (,] y
) )  e.  Con ) )
2322ex 423 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x (,] y ) 
C_  RR  ->  ( y  e.  ( x (,] y )  ->  (
( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( topGen `  ran  (,) )t  ( x (,] y
) )  e.  Con ) ) )
2423com3l 75 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  ( x (,] y )  ->  (
( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( x (,] y )  C_  RR  ->  ( ( topGen `  ran  (,) )t  ( x (,] y
) )  e.  Con ) ) )
2517, 24syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  < 
y )  ->  (
( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( x (,] y )  C_  RR  ->  ( ( topGen `  ran  (,) )t  ( x (,] y
) )  e.  Con ) ) )
26253expia 1153 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  ->  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( x (,] y )  C_  RR  ->  ( ( topGen `  ran  (,) )t  ( x (,] y
) )  e.  Con ) ) ) )
2726pm2.43a 45 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  ->  ( ( x (,] y
)  C_  RR  ->  ( ( topGen `  ran  (,) )t  (
x (,] y ) )  e.  Con )
) )
2827com3l 75 . . . . . . . . . . . . . 14  |-  ( x  <  y  ->  (
( x (,] y
)  C_  RR  ->  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( topGen `  ran  (,) )t  ( x (,] y
) )  e.  Con ) ) )
29 xrlenlt 8890 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  RR*  /\  x  e.  RR* )  ->  (
y  <_  x  <->  -.  x  <  y ) )
3029ancoms 439 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  <_  x  <->  -.  x  <  y ) )
31 ioc0 10703 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( x (,] y
)  =  (/)  <->  y  <_  x ) )
32 retop 18270 . . . . . . . . . . . . . . . . . . . 20  |-  ( topGen ` 
ran  (,) )  e.  Top
33 rest0 16900 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
topGen `  ran  (,) )  e.  Top  ->  ( ( topGen `
 ran  (,) )t  (/) )  =  { (/) } )
34 singempcon 25593 . . . . . . . . . . . . . . . . . . . . 21  |-  { (/) }  e.  Con
3533, 34syl6eqel 2371 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
topGen `  ran  (,) )  e.  Top  ->  ( ( topGen `
 ran  (,) )t  (/) )  e. 
Con )
3632, 35ax-mp 8 . . . . . . . . . . . . . . . . . . 19  |-  ( (
topGen `  ran  (,) )t  (/) )  e. 
Con
3736a1i 10 . . . . . . . . . . . . . . . . . 18  |-  ( ( x (,] y ) 
C_  RR  ->  ( (
topGen `  ran  (,) )t  (/) )  e. 
Con )
38 oveq2 5866 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x (,] y )  =  (/)  ->  ( (
topGen `  ran  (,) )t  (
x (,] y ) )  =  ( (
topGen `  ran  (,) )t  (/) ) )
3938eleq1d 2349 . . . . . . . . . . . . . . . . . 18  |-  ( ( x (,] y )  =  (/)  ->  ( ( ( topGen `  ran  (,) )t  (
x (,] y ) )  e.  Con  <->  ( ( topGen `
 ran  (,) )t  (/) )  e. 
Con ) )
4037, 39syl5ibr 212 . . . . . . . . . . . . . . . . 17  |-  ( ( x (,] y )  =  (/)  ->  ( ( x (,] y ) 
C_  RR  ->  ( (
topGen `  ran  (,) )t  (
x (,] y ) )  e.  Con )
)
4131, 40syl6bir 220 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  <_  x  ->  ( ( x (,] y
)  C_  RR  ->  ( ( topGen `  ran  (,) )t  (
x (,] y ) )  e.  Con )
) )
4230, 41sylbird 226 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  ( -.  x  <  y  -> 
( ( x (,] y )  C_  RR  ->  ( ( topGen `  ran  (,) )t  ( x (,] y
) )  e.  Con ) ) )
4342com3l 75 . . . . . . . . . . . . . 14  |-  ( -.  x  <  y  -> 
( ( x (,] y )  C_  RR  ->  ( ( x  e. 
RR*  /\  y  e.  RR* )  ->  ( ( topGen `
 ran  (,) )t  (
x (,] y ) )  e.  Con )
) )
4428, 43pm2.61i 156 . . . . . . . . . . . . 13  |-  ( ( x (,] y ) 
C_  RR  ->  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( topGen `  ran  (,) )t  (
x (,] y ) )  e.  Con )
)
4516, 44sylbi 187 . . . . . . . . . . . 12  |-  ( ( x (,] y )  e.  ~P RR  ->  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( topGen `  ran  (,) )t  ( x (,] y
) )  e.  Con ) )
4645com12 27 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( x (,] y
)  e.  ~P RR  ->  ( ( topGen `  ran  (,) )t  ( x (,] y
) )  e.  Con ) )
47 eleq1 2343 . . . . . . . . . . . 12  |-  ( I  =  ( x (,] y )  ->  (
I  e.  ~P RR  <->  ( x (,] y )  e.  ~P RR ) )
48 oveq2 5866 . . . . . . . . . . . . 13  |-  ( I  =  ( x (,] y )  ->  (
( topGen `  ran  (,) )t  I
)  =  ( (
topGen `  ran  (,) )t  (
x (,] y ) ) )
4948eleq1d 2349 . . . . . . . . . . . 12  |-  ( I  =  ( x (,] y )  ->  (
( ( topGen `  ran  (,) )t  I )  e.  Con  <->  (
( topGen `  ran  (,) )t  (
x (,] y ) )  e.  Con )
)
5047, 49imbi12d 311 . . . . . . . . . . 11  |-  ( I  =  ( x (,] y )  ->  (
( I  e.  ~P RR  ->  ( ( topGen ` 
ran  (,) )t  I )  e.  Con ) 
<->  ( ( x (,] y )  e.  ~P RR  ->  ( ( topGen ` 
ran  (,) )t  ( x (,] y ) )  e. 
Con ) ) )
5146, 50syl5ibrcom 213 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
I  =  ( x (,] y )  -> 
( I  e.  ~P RR  ->  ( ( topGen ` 
ran  (,) )t  I )  e.  Con ) ) )
5251rexlimivv 2672 . . . . . . . . 9  |-  ( E. x  e.  RR*  E. y  e.  RR*  I  =  ( x (,] y )  ->  ( I  e. 
~P RR  ->  (
( topGen `  ran  (,) )t  I
)  e.  Con )
)
5314, 52sylbi 187 . . . . . . . 8  |-  ( I  e.  ran  (,]  ->  ( I  e.  ~P RR  ->  ( ( topGen `  ran  (,) )t  I )  e.  Con ) )
54 elun 3316 . . . . . . . . 9  |-  ( I  e.  ( ran  [,)  u. 
ran  [,] )  <->  ( I  e.  ran  [,)  \/  I  e.  ran  [,] ) )
55 bsi3 25641 . . . . . . . . . . 11  |-  ( I  e.  ran  [,)  <->  E. x  e.  RR*  E. y  e. 
RR*  I  =  ( x [,) y ) )
56 ovex 5883 . . . . . . . . . . . . . . . 16  |-  ( x [,) y )  e. 
_V
5756elpw 3631 . . . . . . . . . . . . . . 15  |-  ( ( x [,) y )  e.  ~P RR  <->  ( x [,) y )  C_  RR )
58 lbico1 10706 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  < 
y )  ->  x  e.  ( x [,) y
) )
59 ssel2 3175 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( x [,) y
)  C_  RR  /\  x  e.  ( x [,) y
) )  ->  x  e.  RR )
60 icccon2 25700 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( x  e.  RR  /\  y  e.  RR* )  -> 
( ( topGen `  ran  (,) )t  ( x [,) y
) )  e.  Con )
6160ex 423 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  e.  RR  ->  (
y  e.  RR*  ->  ( ( topGen `  ran  (,) )t  (
x [,) y ) )  e.  Con )
)
6261adantld 453 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  e.  RR  ->  (
( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( topGen `  ran  (,) )t  ( x [,) y
) )  e.  Con ) )
6359, 62syl 15 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x [,) y
)  C_  RR  /\  x  e.  ( x [,) y
) )  ->  (
( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( topGen `  ran  (,) )t  ( x [,) y
) )  e.  Con ) )
6463ex 423 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x [,) y ) 
C_  RR  ->  ( x  e.  ( x [,) y )  ->  (
( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( topGen `  ran  (,) )t  ( x [,) y
) )  e.  Con ) ) )
6564com3l 75 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( x [,) y )  ->  (
( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( x [,) y )  C_  RR  ->  ( ( topGen `  ran  (,) )t  ( x [,) y
) )  e.  Con ) ) )
6658, 65syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  < 
y )  ->  (
( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( x [,) y )  C_  RR  ->  ( ( topGen `  ran  (,) )t  ( x [,) y
) )  e.  Con ) ) )
67663expia 1153 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  ->  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( x [,) y )  C_  RR  ->  ( ( topGen `  ran  (,) )t  ( x [,) y
) )  e.  Con ) ) ) )
6867pm2.43a 45 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <  y  ->  ( ( x [,) y
)  C_  RR  ->  ( ( topGen `  ran  (,) )t  (
x [,) y ) )  e.  Con )
) )
6968com3l 75 . . . . . . . . . . . . . . . 16  |-  ( x  <  y  ->  (
( x [,) y
)  C_  RR  ->  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( topGen `  ran  (,) )t  ( x [,) y
) )  e.  Con ) ) )
70 ico0 10702 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( x [,) y
)  =  (/)  <->  y  <_  x ) )
71 oveq2 5866 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x [,) y )  =  (/)  ->  ( (
topGen `  ran  (,) )t  (
x [,) y ) )  =  ( (
topGen `  ran  (,) )t  (/) ) )
7271, 36syl6eqel 2371 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x [,) y )  =  (/)  ->  ( (
topGen `  ran  (,) )t  (
x [,) y ) )  e.  Con )
7372a1d 22 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x [,) y )  =  (/)  ->  ( ( x [,) y ) 
C_  RR  ->  ( (
topGen `  ran  (,) )t  (
x [,) y ) )  e.  Con )
)
7470, 73syl6bir 220 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  <_  x  ->  ( ( x [,) y
)  C_  RR  ->  ( ( topGen `  ran  (,) )t  (
x [,) y ) )  e.  Con )
) )
7530, 74sylbird 226 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  ( -.  x  <  y  -> 
( ( x [,) y )  C_  RR  ->  ( ( topGen `  ran  (,) )t  ( x [,) y
) )  e.  Con ) ) )
7675com3l 75 . . . . . . . . . . . . . . . 16  |-  ( -.  x  <  y  -> 
( ( x [,) y )  C_  RR  ->  ( ( x  e. 
RR*  /\  y  e.  RR* )  ->  ( ( topGen `
 ran  (,) )t  (
x [,) y ) )  e.  Con )
) )
7769, 76pm2.61i 156 . . . . . . . . . . . . . . 15  |-  ( ( x [,) y ) 
C_  RR  ->  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( topGen `  ran  (,) )t  (
x [,) y ) )  e.  Con )
)
7857, 77sylbi 187 . . . . . . . . . . . . . 14  |-  ( ( x [,) y )  e.  ~P RR  ->  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( topGen `  ran  (,) )t  ( x [,) y
) )  e.  Con ) )
7978com12 27 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( x [,) y
)  e.  ~P RR  ->  ( ( topGen `  ran  (,) )t  ( x [,) y
) )  e.  Con ) )
80 eleq1 2343 . . . . . . . . . . . . . 14  |-  ( I  =  ( x [,) y )  ->  (
I  e.  ~P RR  <->  ( x [,) y )  e.  ~P RR ) )
81 oveq2 5866 . . . . . . . . . . . . . . 15  |-  ( I  =  ( x [,) y )  ->  (
( topGen `  ran  (,) )t  I
)  =  ( (
topGen `  ran  (,) )t  (
x [,) y ) ) )
8281eleq1d 2349 . . . . . . . . . . . . . 14  |-  ( I  =  ( x [,) y )  ->  (
( ( topGen `  ran  (,) )t  I )  e.  Con  <->  (
( topGen `  ran  (,) )t  (
x [,) y ) )  e.  Con )
)
8380, 82imbi12d 311 . . . . . . . . . . . . 13  |-  ( I  =  ( x [,) y )  ->  (
( I  e.  ~P RR  ->  ( ( topGen ` 
ran  (,) )t  I )  e.  Con ) 
<->  ( ( x [,) y )  e.  ~P RR  ->  ( ( topGen ` 
ran  (,) )t  ( x [,) y ) )  e. 
Con ) ) )
8479, 83syl5ibrcom 213 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
I  =  ( x [,) y )  -> 
( I  e.  ~P RR  ->  ( ( topGen ` 
ran  (,) )t  I )  e.  Con ) ) )
8584rexlimivv 2672 . . . . . . . . . . 11  |-  ( E. x  e.  RR*  E. y  e.  RR*  I  =  ( x [,) y )  ->  ( I  e. 
~P RR  ->  (
( topGen `  ran  (,) )t  I
)  e.  Con )
)
8655, 85sylbi 187 . . . . . . . . . 10  |-  ( I  e.  ran  [,)  ->  ( I  e.  ~P RR  ->  ( ( topGen `  ran  (,) )t  I )  e.  Con ) )
87 bsi2 25639 . . . . . . . . . . 11  |-  ( I  e.  ran  [,]  <->  E. x  e.  RR*  E. y  e. 
RR*  I  =  ( x [,] y ) )
88 ovex 5883 . . . . . . . . . . . . . . . 16  |-  ( x [,] y )  e. 
_V
8988elpw 3631 . . . . . . . . . . . . . . 15  |-  ( ( x [,] y )  e.  ~P RR  <->  ( x [,] y )  C_  RR )
90 icc0 10704 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( x [,] y
)  =  (/)  <->  y  <  x ) )
91 oveq2 5866 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x [,] y )  =  (/)  ->  ( (
topGen `  ran  (,) )t  (
x [,] y ) )  =  ( (
topGen `  ran  (,) )t  (/) ) )
9291, 36syl6eqel 2371 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x [,] y )  =  (/)  ->  ( (
topGen `  ran  (,) )t  (
x [,] y ) )  e.  Con )
9392a1d 22 . . . . . . . . . . . . . . . . . 18  |-  ( ( x [,] y )  =  (/)  ->  ( ( x [,] y ) 
C_  RR  ->  ( (
topGen `  ran  (,) )t  (
x [,] y ) )  e.  Con )
)
9490, 93syl6bir 220 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
y  <  x  ->  ( ( x [,] y
)  C_  RR  ->  ( ( topGen `  ran  (,) )t  (
x [,] y ) )  e.  Con )
) )
9594com3l 75 . . . . . . . . . . . . . . . 16  |-  ( y  <  x  ->  (
( x [,] y
)  C_  RR  ->  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( topGen `  ran  (,) )t  ( x [,] y
) )  e.  Con ) ) )
96 xrlenlt 8890 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  <->  -.  y  <  x ) )
97 ubicc2 10753 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  <_ 
y )  ->  y  e.  ( x [,] y
) )
98 lbicc2 10752 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  <_ 
y )  ->  x  e.  ( x [,] y
) )
99 ssel2 3175 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( x [,] y
)  C_  RR  /\  y  e.  ( x [,] y
) )  ->  y  e.  RR )
100 ssel2 3175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( x [,] y
)  C_  RR  /\  x  e.  ( x [,] y
) )  ->  x  e.  RR )
101 iccconn 18335 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( x [,] y
) )  e.  Con )
102101ex 423 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( x  e.  RR  ->  (
y  e.  RR  ->  ( ( topGen `  ran  (,) )t  (
x [,] y ) )  e.  Con )
)
103100, 102syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( x [,] y
)  C_  RR  /\  x  e.  ( x [,] y
) )  ->  (
y  e.  RR  ->  ( ( topGen `  ran  (,) )t  (
x [,] y ) )  e.  Con )
)
104103ex 423 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( x [,] y ) 
C_  RR  ->  ( x  e.  ( x [,] y )  ->  (
y  e.  RR  ->  ( ( topGen `  ran  (,) )t  (
x [,] y ) )  e.  Con )
) )
105104com3r 73 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( y  e.  RR  ->  (
( x [,] y
)  C_  RR  ->  ( x  e.  ( x [,] y )  -> 
( ( topGen `  ran  (,) )t  ( x [,] y
) )  e.  Con ) ) )
10699, 105syl 15 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( x [,] y
)  C_  RR  /\  y  e.  ( x [,] y
) )  ->  (
( x [,] y
)  C_  RR  ->  ( x  e.  ( x [,] y )  -> 
( ( topGen `  ran  (,) )t  ( x [,] y
) )  e.  Con ) ) )
107106ex 423 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x [,] y ) 
C_  RR  ->  ( y  e.  ( x [,] y )  ->  (
( x [,] y
)  C_  RR  ->  ( x  e.  ( x [,] y )  -> 
( ( topGen `  ran  (,) )t  ( x [,] y
) )  e.  Con ) ) ) )
108107pm2.43a 45 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x [,] y ) 
C_  RR  ->  ( y  e.  ( x [,] y )  ->  (
x  e.  ( x [,] y )  -> 
( ( topGen `  ran  (,) )t  ( x [,] y
) )  e.  Con ) ) )
109108com3l 75 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  ( x [,] y )  ->  (
x  e.  ( x [,] y )  -> 
( ( x [,] y )  C_  RR  ->  ( ( topGen `  ran  (,) )t  ( x [,] y
) )  e.  Con ) ) )
110109a1dd 42 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  ( x [,] y )  ->  (
x  e.  ( x [,] y )  -> 
( ( x  e. 
RR*  /\  y  e.  RR* )  ->  ( (
x [,] y ) 
C_  RR  ->  ( (
topGen `  ran  (,) )t  (
x [,] y ) )  e.  Con )
) ) )
11197, 98, 110sylc 56 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  <_ 
y )  ->  (
( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( x [,] y )  C_  RR  ->  ( ( topGen `  ran  (,) )t  ( x [,] y
) )  e.  Con ) ) )
1121113expia 1153 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  ->  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( x [,] y )  C_  RR  ->  ( ( topGen `  ran  (,) )t  ( x [,] y
) )  e.  Con ) ) ) )
113112pm2.43a 45 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x  <_  y  ->  ( ( x [,] y
)  C_  RR  ->  ( ( topGen `  ran  (,) )t  (
x [,] y ) )  e.  Con )
) )
11496, 113sylbird 226 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  ( -.  y  <  x  -> 
( ( x [,] y )  C_  RR  ->  ( ( topGen `  ran  (,) )t  ( x [,] y
) )  e.  Con ) ) )
115114com3l 75 . . . . . . . . . . . . . . . 16  |-  ( -.  y  <  x  -> 
( ( x [,] y )  C_  RR  ->  ( ( x  e. 
RR*  /\  y  e.  RR* )  ->  ( ( topGen `
 ran  (,) )t  (
x [,] y ) )  e.  Con )
) )
11695, 115pm2.61i 156 . . . . . . . . . . . . . . 15  |-  ( ( x [,] y ) 
C_  RR  ->  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( topGen `  ran  (,) )t  (
x [,] y ) )  e.  Con )
)
11789, 116sylbi 187 . . . . . . . . . . . . . 14  |-  ( ( x [,] y )  e.  ~P RR  ->  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  ( ( topGen `  ran  (,) )t  ( x [,] y
) )  e.  Con ) )
118117com12 27 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
( x [,] y
)  e.  ~P RR  ->  ( ( topGen `  ran  (,) )t  ( x [,] y
) )  e.  Con ) )
119 eleq1 2343 . . . . . . . . . . . . . 14  |-  ( I  =  ( x [,] y )  ->  (
I  e.  ~P RR  <->  ( x [,] y )  e.  ~P RR ) )
120 oveq2 5866 . . . . . . . . . . . . . . 15  |-  ( I  =  ( x [,] y )  ->  (
( topGen `  ran  (,) )t  I
)  =  ( (
topGen `  ran  (,) )t  (
x [,] y ) ) )
121120eleq1d 2349 . . . . . . . . . . . . . 14  |-  ( I  =  ( x [,] y )  ->  (
( ( topGen `  ran  (,) )t  I )  e.  Con  <->  (
( topGen `  ran  (,) )t  (
x [,] y ) )  e.  Con )
)
122119, 121imbi12d 311 . . . . . . . . . . . . 13  |-  ( I  =  ( x [,] y )  ->  (
( I  e.  ~P RR  ->  ( ( topGen ` 
ran  (,) )t  I )  e.  Con ) 
<->  ( ( x [,] y )  e.  ~P RR  ->  ( ( topGen ` 
ran  (,) )t  ( x [,] y ) )  e. 
Con ) ) )
123118, 122syl5ibrcom 213 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
I  =  ( x [,] y )  -> 
( I  e.  ~P RR  ->  ( ( topGen ` 
ran  (,) )t  I )  e.  Con ) ) )
124123rexlimivv 2672 . . . . . . . . . . 11  |-  ( E. x  e.  RR*  E. y  e.  RR*  I  =  ( x [,] y )  ->  ( I  e. 
~P RR  ->  (
( topGen `  ran  (,) )t  I
)  e.  Con )
)
12587, 124sylbi 187 . . . . . . . . . 10  |-  ( I  e.  ran  [,]  ->  ( I  e.  ~P RR  ->  ( ( topGen `  ran  (,) )t  I )  e.  Con ) )
12686, 125jaoi 368 . . . . . . . . 9  |-  ( ( I  e.  ran  [,)  \/  I  e.  ran  [,] )  ->  ( I  e. 
~P RR  ->  (
( topGen `  ran  (,) )t  I
)  e.  Con )
)
12754, 126sylbi 187 . . . . . . . 8  |-  ( I  e.  ( ran  [,)  u. 
ran  [,] )  ->  (
I  e.  ~P RR  ->  ( ( topGen `  ran  (,) )t  I )  e.  Con ) )
12853, 127jaoi 368 . . . . . . 7  |-  ( ( I  e.  ran  (,]  \/  I  e.  ( ran 
[,)  u.  ran  [,] )
)  ->  ( I  e.  ~P RR  ->  (
( topGen `  ran  (,) )t  I
)  e.  Con )
)
12913, 128sylbi 187 . . . . . 6  |-  ( I  e.  ( ran  (,]  u.  ( ran  [,)  u.  ran  [,] ) )  -> 
( I  e.  ~P RR  ->  ( ( topGen ` 
ran  (,) )t  I )  e.  Con ) )
13012, 129jaoi 368 . . . . 5  |-  ( ( I  e.  ran  (,)  \/  I  e.  ( ran 
(,]  u.  ( ran  [,) 
u.  ran  [,] )
) )  ->  (
I  e.  ~P RR  ->  ( ( topGen `  ran  (,) )t  I )  e.  Con ) )
1312, 130sylbi 187 . . . 4  |-  ( I  e.  ( ran  (,)  u.  ( ran  (,]  u.  ( ran  [,)  u.  ran  [,] ) ) )  -> 
( I  e.  ~P RR  ->  ( ( topGen ` 
ran  (,) )t  I )  e.  Con ) )
132131imp 418 . . 3  |-  ( ( I  e.  ( ran 
(,)  u.  ( ran  (,] 
u.  ( ran  [,)  u. 
ran  [,] ) ) )  /\  I  e.  ~P RR )  ->  ( (
topGen `  ran  (,) )t  I
)  e.  Con )
1331, 132sylbi 187 . 2  |-  ( I  e.  ( ( ran 
(,)  u.  ( ran  (,] 
u.  ( ran  [,)  u. 
ran  [,] ) ) )  i^i  ~P RR )  ->  ( ( topGen ` 
ran  (,) )t  I )  e.  Con )
134 df-intvl 25697 . 2  |-  Intvl  =  ( ( ran  (,)  u.  ( ran  (,]  u.  ( ran  [,)  u.  ran  [,] ) ) )  i^i 
~P RR )
135133, 134eleq2s 2375 1  |-  ( I  e.  Intvl  ->  ( ( topGen `
 ran  (,) )t  I
)  e.  Con )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   class class class wbr 4023   ran crn 4690   ` cfv 5255  (class class class)co 5858   RRcr 8736   RR*cxr 8866    < clt 8867    <_ cle 8868   (,)cioo 10656   (,]cioc 10657   [,)cico 10658   [,]cicc 10659   ↾t crest 13325   topGenctg 13342   Topctop 16631   Conccon 17137   Intvlcintvl 25696
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-rest 13327  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-con 17138  df-intvl 25697
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