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Theorem opnreen 21835
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 9630 . . . . 5  |-  RR  e.  _V
2 elssuni 4245 . . . . . 6  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  U. ( topGen `
 ran  (,) )
)
3 uniretop 21769 . . . . . 6  |-  RR  =  U. ( topGen `  ran  (,) )
42, 3syl6sseqr 3511 . . . . 5  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  RR )
5 ssdomg 7618 . . . . 5  |-  ( RR  e.  _V  ->  ( A  C_  RR  ->  A  ~<_  RR ) )
61, 4, 5mpsyl 65 . . . 4  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  ~<_  RR )
7 rpnnen 14266 . . . 4  |-  RR  ~~  ~P NN
8 domentr 7631 . . . 4  |-  ( ( A  ~<_  RR  /\  RR  ~~  ~P NN )  ->  A  ~<_  ~P NN )
96, 7, 8sylancl 666 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  ~<_  ~P NN )
109adantr 466 . 2  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  A  =/=  (/) )  ->  A  ~<_  ~P NN )
11 n0 3771 . . . 4  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
124sselda 3464 . . . . . . . . . 10  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  x  e.  A )  ->  x  e.  RR )
13 eqid 2422 . . . . . . . . . . . . . 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 14265 . . . . . . . . . . . . 13  |-  ~P NN  ~<_  ( 0 [,] 1
)
15 rphalfcl 11327 . . . . . . . . . . . . . . . 16  |-  ( y  e.  RR+  ->  ( y  /  2 )  e.  RR+ )
1615rpred 11341 . . . . . . . . . . . . . . 15  |-  ( y  e.  RR+  ->  ( y  /  2 )  e.  RR )
17 resubcl 9938 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  ( y  /  2
)  e.  RR )  ->  ( x  -  ( y  /  2
) )  e.  RR )
1816, 17sylan2 476 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  -  (
y  /  2 ) )  e.  RR )
19 readdcl 9622 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  ( y  /  2
)  e.  RR )  ->  ( x  +  ( y  /  2
) )  e.  RR )
2016, 19sylan2 476 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  +  ( y  /  2 ) )  e.  RR )
21 simpl 458 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  ->  x  e.  RR )
22 ltsubrp 11335 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  ( y  /  2
)  e.  RR+ )  ->  ( x  -  (
y  /  2 ) )  <  x )
2315, 22sylan2 476 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  -  (
y  /  2 ) )  <  x )
24 ltaddrp 11336 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  ( y  /  2
)  e.  RR+ )  ->  x  <  ( x  +  ( y  / 
2 ) ) )
2515, 24sylan2 476 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  ->  x  <  ( x  +  ( y  /  2
) ) )
2618, 21, 20, 23, 25lttrd 9796 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  -  (
y  /  2 ) )  <  ( x  +  ( y  / 
2 ) ) )
27 iccen 11777 . . . . . . . . . . . . . 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 1264 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( 0 [,] 1
)  ~~  ( (
x  -  ( y  /  2 ) ) [,] ( x  +  ( y  /  2
) ) ) )
29 domentr 7631 . . . . . . . . . . . . 13  |-  ( ( ~P NN  ~<_  ( 0 [,] 1 )  /\  ( 0 [,] 1
)  ~~  ( (
x  -  ( y  /  2 ) ) [,] ( x  +  ( y  /  2
) ) ) )  ->  ~P NN  ~<_  ( ( x  -  ( y  /  2 ) ) [,] ( x  +  ( y  /  2
) ) ) )
3014, 28, 29sylancr 667 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  ->  ~P NN  ~<_  ( ( x  -  ( y  / 
2 ) ) [,] ( x  +  ( y  /  2 ) ) ) )
31 ovex 6329 . . . . . . . . . . . . 13  |-  ( ( x  -  y ) (,) ( x  +  y ) )  e. 
_V
32 rpre 11308 . . . . . . . . . . . . . . . 16  |-  ( y  e.  RR+  ->  y  e.  RR )
33 resubcl 9938 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  -  y
)  e.  RR )
3432, 33sylan2 476 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  -  y
)  e.  RR )
3534rexrd 9690 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  -  y
)  e.  RR* )
36 readdcl 9622 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
3732, 36sylan2 476 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  +  y )  e.  RR )
3837rexrd 9690 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  +  y )  e.  RR* )
3921recnd 9669 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  ->  x  e.  CC )
4016adantl 467 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( y  /  2
)  e.  RR )
4140recnd 9669 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( y  /  2
)  e.  CC )
4239, 41, 41subsub4d 10017 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( x  -  ( y  /  2
) )  -  (
y  /  2 ) )  =  ( x  -  ( ( y  /  2 )  +  ( y  /  2
) ) ) )
4332adantl 467 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
y  e.  RR )
4443recnd 9669 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
y  e.  CC )
45442halvesd 10858 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( y  / 
2 )  +  ( y  /  2 ) )  =  y )
4645oveq2d 6317 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  -  (
( y  /  2
)  +  ( y  /  2 ) ) )  =  ( x  -  y ) )
4742, 46eqtrd 2463 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( x  -  ( y  /  2
) )  -  (
y  /  2 ) )  =  ( x  -  y ) )
4815adantl 467 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( y  /  2
)  e.  RR+ )
4918, 48ltsubrpd 11370 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( x  -  ( y  /  2
) )  -  (
y  /  2 ) )  <  ( x  -  ( y  / 
2 ) ) )
5047, 49eqbrtrrd 4443 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  -  y
)  <  ( x  -  ( y  / 
2 ) ) )
51 ltaddrp 11336 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  +  ( y  /  2 ) )  e.  RR  /\  ( y  /  2
)  e.  RR+ )  ->  ( x  +  ( y  /  2 ) )  <  ( ( x  +  ( y  /  2 ) )  +  ( y  / 
2 ) ) )
5220, 48, 51syl2anc 665 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  +  ( y  /  2 ) )  <  ( ( x  +  ( y  /  2 ) )  +  ( y  / 
2 ) ) )
5339, 41, 41addassd 9665 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( x  +  ( y  /  2
) )  +  ( y  /  2 ) )  =  ( x  +  ( ( y  /  2 )  +  ( y  /  2
) ) ) )
5445oveq2d 6317 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  +  ( ( y  /  2
)  +  ( y  /  2 ) ) )  =  ( x  +  y ) )
5553, 54eqtrd 2463 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( x  +  ( y  /  2
) )  +  ( y  /  2 ) )  =  ( x  +  y ) )
5652, 55breqtrd 4445 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x  +  ( y  /  2 ) )  <  ( x  +  y ) )
57 iccssioo 11703 . . . . . . . . . . . . . 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 1265 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( x  -  ( y  /  2
) ) [,] (
x  +  ( y  /  2 ) ) )  C_  ( (
x  -  y ) (,) ( x  +  y ) ) )
59 ssdomg 7618 . . . . . . . . . . . . 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 65 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( ( x  -  ( y  /  2
) ) [,] (
x  +  ( y  /  2 ) ) )  ~<_  ( ( x  -  y ) (,) ( x  +  y ) ) )
61 domtr 7625 . . . . . . . . . . . 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 665 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  ->  ~P NN  ~<_  ( ( x  -  y ) (,) ( x  +  y ) ) )
63 eqid 2422 . . . . . . . . . . . . 13  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
6463bl2ioo 21796 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) y )  =  ( ( x  -  y ) (,) (
x  +  y ) ) )
6532, 64sylan2 476 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  -> 
( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) y )  =  ( ( x  -  y ) (,) (
x  +  y ) ) )
6662, 65breqtrrd 4447 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  RR+ )  ->  ~P NN  ~<_  ( x (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) y ) )
6712, 66sylan 473 . . . . . . . . 9  |-  ( ( ( A  e.  (
topGen `  ran  (,) )  /\  x  e.  A
)  /\  y  e.  RR+ )  ->  ~P NN  ~<_  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) y ) )
6867adantr 466 . . . . . . . 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 766 . . . . . . . . 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 462 . . . . . . . . 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 7618 . . . . . . . . 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 62 . . . . . . . 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 7625 . . . . . . . 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 665 . . . . . . 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 2422 . . . . . . . . . 10  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
7663, 75tgioo 21800 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
7776eleq2i 2500 . . . . . . . 8  |-  ( A  e.  ( topGen `  ran  (,) )  <->  A  e.  ( MetOpen
`  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) ) )
7863rexmet 21795 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )
7975mopni2 21494 . . . . . . . . 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 1347 . . . . . . . 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 474 . . . . . . 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 2970 . . . . . 6  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  x  e.  A )  ->  ~P NN 
~<_  A )
8382ex 435 . . . . 5  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( x  e.  A  ->  ~P NN  ~<_  A ) )
8483exlimdv 1768 . . . 4  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( E. x  x  e.  A  ->  ~P NN  ~<_  A ) )
8511, 84syl5bi 220 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( A  =/=  (/)  ->  ~P NN  ~<_  A ) )
8685imp 430 . 2  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  A  =/=  (/) )  ->  ~P NN 
~<_  A )
87 sbth 7694 . 2  |-  ( ( A  ~<_  ~P NN  /\  ~P NN 
~<_  A )  ->  A  ~~  ~P NN )
8810, 86, 87syl2anc 665 1  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  A  =/=  (/) )  ->  A  ~~  ~P NN )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1868    =/= wne 2618   E.wrex 2776   _Vcvv 3081    C_ wss 3436   (/)c0 3761   ifcif 3909   ~Pcpw 3979   U.cuni 4216   class class class wbr 4420    |-> cmpt 4479    X. cxp 4847   ran crn 4850    |` cres 4851    o. ccom 4853   ` cfv 5597  (class class class)co 6301    ~~ cen 7570    ~<_ cdom 7571   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542   RR*cxr 9674    < clt 9675    - cmin 9860    / cdiv 10269   NNcn 10609   2c2 10659   3c3 10660   RR+crp 11302   (,)cioo 11635   [,]cicc 11638   ^cexp 12271   abscabs 13285   topGenctg 15323   *Metcxmt 18942   ballcbl 18944   MetOpencmopn 18947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-omul 7191  df-er 7367  df-map 7478  df-pm 7479  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-acn 8377  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12027  df-seq 12213  df-exp 12272  df-hash 12515  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-limsup 13513  df-clim 13539  df-rlim 13540  df-sum 13740  df-topgen 15329  df-psmet 18949  df-xmet 18950  df-met 18951  df-bl 18952  df-mopn 18953  df-top 19907  df-bases 19908  df-topon 19909
This theorem is referenced by:  rectbntr0  21836
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