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Theorem opnreen 21462
Description: Every nonempty open set is uncountable. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 20-Feb-2015.)
Assertion
Ref Expression
opnreen  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  A  =/=  (/) )  ->  A  ~~  ~P NN )

Proof of Theorem opnreen
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reex 9600 . . . . 5  |-  RR  e.  _V
2 elssuni 4281 . . . . . 6  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  U. ( topGen `
 ran  (,) )
)
3 uniretop 21395 . . . . . 6  |-  RR  =  U. ( topGen `  ran  (,) )
42, 3syl6sseqr 3546 . . . . 5  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  RR )
5 ssdomg 7580 . . . . 5  |-  ( RR  e.  _V  ->  ( A  C_  RR  ->  A  ~<_  RR ) )
61, 4, 5mpsyl 63 . . . 4  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  ~<_  RR )
7 rpnnen 13972 . . . 4  |-  RR  ~~  ~P NN
8 domentr 7593 . . . 4  |-  ( ( A  ~<_  RR  /\  RR  ~~  ~P NN )  ->  A  ~<_  ~P NN )
96, 7, 8sylancl 662 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  ~<_  ~P NN )
109adantr 465 . 2  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  A  =/=  (/) )  ->  A  ~<_  ~P NN )
11 n0 3803 . . . 4  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
124sselda 3499 . . . . . . . . . 10  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  x  e.  A )  ->  x  e.  RR )
13 eqid 2457 . . . . . . . . . . . . . 14  |-  ( x  e.  ~P NN  |->  ( y  e.  NN  |->  if ( y  e.  x ,  ( ( 1  /  3 ) ^
y ) ,  0 ) ) )  =  ( x  e.  ~P NN  |->  ( y  e.  NN  |->  if ( y  e.  x ,  ( ( 1  /  3
) ^ y ) ,  0 ) ) )
1413rpnnen2 13971 . . . . . . . . . . . . 13  |-  ~P NN  ~<_  ( 0 [,] 1
)
15 rphalfcl 11269 . . . . . . . . . . . . . . . 16  |-  ( y  e.  RR+  ->  ( y  /  2 )  e.  RR+ )
1615rpred 11281 . . . . . . . . . . . . . . 15  |-  ( y  e.  RR+  ->  ( y  /  2 )  e.  RR )
17 resubcl 9902 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  ( y  /  2
)  e.  RR )  ->  ( x  -  ( y  /  2
) )  e.  RR )
1816, 17sylan2 474 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  -  (
y  /  2 ) )  e.  RR )
19 readdcl 9592 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  ( y  /  2
)  e.  RR )  ->  ( x  +  ( y  /  2
) )  e.  RR )
2016, 19sylan2 474 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  +  ( y  /  2 ) )  e.  RR )
21 simpl 457 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  ->  x  e.  RR )
22 ltsubrp 11276 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  ( y  /  2
)  e.  RR+ )  ->  ( x  -  (
y  /  2 ) )  <  x )
2315, 22sylan2 474 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  -  (
y  /  2 ) )  <  x )
24 ltaddrp 11277 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  ( y  /  2
)  e.  RR+ )  ->  x  <  ( x  +  ( y  / 
2 ) ) )
2515, 24sylan2 474 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  ->  x  <  ( x  +  ( y  /  2
) ) )
2618, 21, 20, 23, 25lttrd 9760 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  -  (
y  /  2 ) )  <  ( x  +  ( y  / 
2 ) ) )
27 iccen 11690 . . . . . . . . . . . . . 14  |-  ( ( ( x  -  (
y  /  2 ) )  e.  RR  /\  ( x  +  (
y  /  2 ) )  e.  RR  /\  ( x  -  (
y  /  2 ) )  <  ( x  +  ( y  / 
2 ) ) )  ->  ( 0 [,] 1 )  ~~  (
( x  -  (
y  /  2 ) ) [,] ( x  +  ( y  / 
2 ) ) ) )
2818, 20, 26, 27syl3anc 1228 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( 0 [,] 1
)  ~~  ( (
x  -  ( y  /  2 ) ) [,] ( x  +  ( y  /  2
) ) ) )
29 domentr 7593 . . . . . . . . . . . . 13  |-  ( ( ~P NN  ~<_  ( 0 [,] 1 )  /\  ( 0 [,] 1
)  ~~  ( (
x  -  ( y  /  2 ) ) [,] ( x  +  ( y  /  2
) ) ) )  ->  ~P NN  ~<_  ( ( x  -  ( y  /  2 ) ) [,] ( x  +  ( y  /  2
) ) ) )
3014, 28, 29sylancr 663 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  ->  ~P NN  ~<_  ( ( x  -  ( y  / 
2 ) ) [,] ( x  +  ( y  /  2 ) ) ) )
31 ovex 6324 . . . . . . . . . . . . 13  |-  ( ( x  -  y ) (,) ( x  +  y ) )  e. 
_V
32 rpre 11251 . . . . . . . . . . . . . . . 16  |-  ( y  e.  RR+  ->  y  e.  RR )
33 resubcl 9902 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  -  y
)  e.  RR )
3432, 33sylan2 474 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  -  y
)  e.  RR )
3534rexrd 9660 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  -  y
)  e.  RR* )
36 readdcl 9592 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
3732, 36sylan2 474 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  +  y )  e.  RR )
3837rexrd 9660 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  +  y )  e.  RR* )
3921recnd 9639 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  ->  x  e.  CC )
4016adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( y  /  2
)  e.  RR )
4140recnd 9639 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( y  /  2
)  e.  CC )
4239, 41, 41subsub4d 9981 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( x  -  ( y  /  2
) )  -  (
y  /  2 ) )  =  ( x  -  ( ( y  /  2 )  +  ( y  /  2
) ) ) )
4332adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
y  e.  RR )
4443recnd 9639 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
y  e.  CC )
45442halvesd 10805 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( y  / 
2 )  +  ( y  /  2 ) )  =  y )
4645oveq2d 6312 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  -  (
( y  /  2
)  +  ( y  /  2 ) ) )  =  ( x  -  y ) )
4742, 46eqtrd 2498 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( x  -  ( y  /  2
) )  -  (
y  /  2 ) )  =  ( x  -  y ) )
4815adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( y  /  2
)  e.  RR+ )
4918, 48ltsubrpd 11309 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( x  -  ( y  /  2
) )  -  (
y  /  2 ) )  <  ( x  -  ( y  / 
2 ) ) )
5047, 49eqbrtrrd 4478 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  -  y
)  <  ( x  -  ( y  / 
2 ) ) )
51 ltaddrp 11277 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  +  ( y  /  2 ) )  e.  RR  /\  ( y  /  2
)  e.  RR+ )  ->  ( x  +  ( y  /  2 ) )  <  ( ( x  +  ( y  /  2 ) )  +  ( y  / 
2 ) ) )
5220, 48, 51syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  +  ( y  /  2 ) )  <  ( ( x  +  ( y  /  2 ) )  +  ( y  / 
2 ) ) )
5339, 41, 41addassd 9635 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( x  +  ( y  /  2
) )  +  ( y  /  2 ) )  =  ( x  +  ( ( y  /  2 )  +  ( y  /  2
) ) ) )
5445oveq2d 6312 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  +  ( ( y  /  2
)  +  ( y  /  2 ) ) )  =  ( x  +  y ) )
5553, 54eqtrd 2498 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( x  +  ( y  /  2
) )  +  ( y  /  2 ) )  =  ( x  +  y ) )
5652, 55breqtrd 4480 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  +  ( y  /  2 ) )  <  ( x  +  y ) )
57 iccssioo 11618 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  -  y )  e.  RR*  /\  ( x  +  y )  e.  RR* )  /\  ( ( x  -  y )  <  (
x  -  ( y  /  2 ) )  /\  ( x  +  ( y  /  2
) )  <  (
x  +  y ) ) )  ->  (
( x  -  (
y  /  2 ) ) [,] ( x  +  ( y  / 
2 ) ) ) 
C_  ( ( x  -  y ) (,) ( x  +  y ) ) )
5835, 38, 50, 56, 57syl22anc 1229 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( x  -  ( y  /  2
) ) [,] (
x  +  ( y  /  2 ) ) )  C_  ( (
x  -  y ) (,) ( x  +  y ) ) )
59 ssdomg 7580 . . . . . . . . . . . . 13  |-  ( ( ( x  -  y
) (,) ( x  +  y ) )  e.  _V  ->  (
( ( x  -  ( y  /  2
) ) [,] (
x  +  ( y  /  2 ) ) )  C_  ( (
x  -  y ) (,) ( x  +  y ) )  -> 
( ( x  -  ( y  /  2
) ) [,] (
x  +  ( y  /  2 ) ) )  ~<_  ( ( x  -  y ) (,) ( x  +  y ) ) ) )
6031, 58, 59mpsyl 63 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( x  -  ( y  /  2
) ) [,] (
x  +  ( y  /  2 ) ) )  ~<_  ( ( x  -  y ) (,) ( x  +  y ) ) )
61 domtr 7587 . . . . . . . . . . . 12  |-  ( ( ~P NN  ~<_  ( ( x  -  ( y  /  2 ) ) [,] ( x  +  ( y  /  2
) ) )  /\  ( ( x  -  ( y  /  2
) ) [,] (
x  +  ( y  /  2 ) ) )  ~<_  ( ( x  -  y ) (,) ( x  +  y ) ) )  ->  ~P NN  ~<_  ( ( x  -  y ) (,) ( x  +  y ) ) )
6230, 60, 61syl2anc 661 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  ->  ~P NN  ~<_  ( ( x  -  y ) (,) ( x  +  y ) ) )
63 eqid 2457 . . . . . . . . . . . . 13  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
6463bl2ioo 21423 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) y )  =  ( ( x  -  y ) (,) (
x  +  y ) ) )
6532, 64sylan2 474 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) y )  =  ( ( x  -  y ) (,) (
x  +  y ) ) )
6662, 65breqtrrd 4482 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  ->  ~P NN  ~<_  ( x (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) y ) )
6712, 66sylan 471 . . . . . . . . 9  |-  ( ( ( A  e.  (
topGen `  ran  (,) )  /\  x  e.  A
)  /\  y  e.  RR+ )  ->  ~P NN  ~<_  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) y ) )
6867adantr 465 . . . . . . . 8  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  x  e.  A
)  /\  y  e.  RR+ )  /\  ( x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) y )  C_  A )  ->  ~P NN  ~<_  ( x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) y ) )
69 simplll 759 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  x  e.  A
)  /\  y  e.  RR+ )  /\  ( x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) y )  C_  A )  ->  A  e.  ( topGen ` 
ran  (,) ) )
70 simpr 461 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  x  e.  A
)  /\  y  e.  RR+ )  /\  ( x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) y )  C_  A )  ->  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) y )  C_  A )
71 ssdomg 7580 . . . . . . . . 9  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( (
x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) y )  C_  A  ->  ( x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) y )  ~<_  A ) )
7269, 70, 71sylc 60 . . . . . . . 8  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  x  e.  A
)  /\  y  e.  RR+ )  /\  ( x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) y )  C_  A )  ->  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) y )  ~<_  A )
73 domtr 7587 . . . . . . . 8  |-  ( ( ~P NN  ~<_  ( x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) y )  /\  ( x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) y )  ~<_  A )  ->  ~P NN  ~<_  A )
7468, 72, 73syl2anc 661 . . . . . . 7  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  x  e.  A
)  /\  y  e.  RR+ )  /\  ( x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) y )  C_  A )  ->  ~P NN  ~<_  A )
75 eqid 2457 . . . . . . . . . 10  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
7663, 75tgioo 21427 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
7776eleq2i 2535 . . . . . . . 8  |-  ( A  e.  ( topGen `  ran  (,) )  <->  A  e.  ( MetOpen
`  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) ) )
7863rexmet 21422 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )
7975mopni2 21122 . . . . . . . . 9  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( *Met `  RR )  /\  A  e.  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )  /\  x  e.  A )  ->  E. y  e.  RR+  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) y )  C_  A )
8078, 79mp3an1 1311 . . . . . . . 8  |-  ( ( A  e.  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  /\  x  e.  A )  ->  E. y  e.  RR+  ( x (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) y )  C_  A )
8177, 80sylanb 472 . . . . . . 7  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  x  e.  A )  ->  E. y  e.  RR+  ( x (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) y )  C_  A )
8274, 81r19.29a 2999 . . . . . 6  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  x  e.  A )  ->  ~P NN 
~<_  A )
8382ex 434 . . . . 5  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( x  e.  A  ->  ~P NN  ~<_  A ) )
8483exlimdv 1725 . . . 4  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( E. x  x  e.  A  ->  ~P NN  ~<_  A ) )
8511, 84syl5bi 217 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( A  =/=  (/)  ->  ~P NN  ~<_  A ) )
8685imp 429 . 2  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  A  =/=  (/) )  ->  ~P NN 
~<_  A )
87 sbth 7656 . 2  |-  ( ( A  ~<_  ~P NN  /\  ~P NN 
~<_  A )  ->  A  ~~  ~P NN )
8810, 86, 87syl2anc 661 1  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  A  =/=  (/) )  ->  A  ~~  ~P NN )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   E.wrex 2808   _Vcvv 3109    C_ wss 3471   (/)c0 3793   ifcif 3944   ~Pcpw 4015   U.cuni 4251   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   ran crn 5009    |` cres 5010    o. ccom 5012   ` cfv 5594  (class class class)co 6296    ~~ cen 7532    ~<_ cdom 7533   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512   RR*cxr 9644    < clt 9645    - cmin 9824    / cdiv 10227   NNcn 10556   2c2 10606   3c3 10607   RR+crp 11245   (,)cioo 11554   [,]cicc 11557   ^cexp 12169   abscabs 13079   topGenctg 14855   *Metcxmt 18530   ballcbl 18532   MetOpencmopn 18535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-top 19526  df-bases 19528  df-topon 19529
This theorem is referenced by:  rectbntr0  21463
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