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Theorem supnfcls 17731
Description: The filter of supersets of  X  \  U does not cluster at any point of the open set  U. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
supnfcls  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  ->  -.  A  e.  ( J  fClus  { x  e.  ~P X  |  ( X  \  U )  C_  x } ) )
Distinct variable groups:    x, J    x, X    x, U
Allowed substitution hint:    A( x)

Proof of Theorem supnfcls
StepHypRef Expression
1 disjdif 3539 . 2  |-  ( U  i^i  ( X  \  U ) )  =  (/)
2 simpr 447 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  /\  A  e.  ( J  fClus  { x  e.  ~P X  |  ( X  \  U ) 
C_  x } ) )  ->  A  e.  ( J  fClus  { x  e.  ~P X  |  ( X  \  U ) 
C_  x } ) )
3 simpl2 959 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  /\  A  e.  ( J  fClus  { x  e.  ~P X  |  ( X  \  U ) 
C_  x } ) )  ->  U  e.  J )
4 simpl3 960 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  /\  A  e.  ( J  fClus  { x  e.  ~P X  |  ( X  \  U ) 
C_  x } ) )  ->  A  e.  U )
5 difss 3316 . . . . . . 7  |-  ( X 
\  U )  C_  X
6 simpl1 958 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  /\  A  e.  ( J  fClus  { x  e.  ~P X  |  ( X  \  U ) 
C_  x } ) )  ->  J  e.  (TopOn `  X ) )
7 toponmax 16682 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
8 elpw2g 4190 . . . . . . . 8  |-  ( X  e.  J  ->  (
( X  \  U
)  e.  ~P X  <->  ( X  \  U ) 
C_  X ) )
96, 7, 83syl 18 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  /\  A  e.  ( J  fClus  { x  e.  ~P X  |  ( X  \  U ) 
C_  x } ) )  ->  ( ( X  \  U )  e. 
~P X  <->  ( X  \  U )  C_  X
) )
105, 9mpbiri 224 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  /\  A  e.  ( J  fClus  { x  e.  ~P X  |  ( X  \  U ) 
C_  x } ) )  ->  ( X  \  U )  e.  ~P X )
11 ssid 3210 . . . . . . 7  |-  ( X 
\  U )  C_  ( X  \  U )
1211a1i 10 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  /\  A  e.  ( J  fClus  { x  e.  ~P X  |  ( X  \  U ) 
C_  x } ) )  ->  ( X  \  U )  C_  ( X  \  U ) )
13 sseq2 3213 . . . . . . 7  |-  ( x  =  ( X  \  U )  ->  (
( X  \  U
)  C_  x  <->  ( X  \  U )  C_  ( X  \  U ) ) )
1413elrab 2936 . . . . . 6  |-  ( ( X  \  U )  e.  { x  e. 
~P X  |  ( X  \  U ) 
C_  x }  <->  ( ( X  \  U )  e. 
~P X  /\  ( X  \  U )  C_  ( X  \  U ) ) )
1510, 12, 14sylanbrc 645 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  /\  A  e.  ( J  fClus  { x  e.  ~P X  |  ( X  \  U ) 
C_  x } ) )  ->  ( X  \  U )  e.  {
x  e.  ~P X  |  ( X  \  U )  C_  x } )
16 fclsopni 17726 . . . . 5  |-  ( ( A  e.  ( J 
fClus  { x  e.  ~P X  |  ( X  \  U )  C_  x } )  /\  ( U  e.  J  /\  A  e.  U  /\  ( X  \  U )  e.  { x  e. 
~P X  |  ( X  \  U ) 
C_  x } ) )  ->  ( U  i^i  ( X  \  U
) )  =/=  (/) )
172, 3, 4, 15, 16syl13anc 1184 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  /\  A  e.  ( J  fClus  { x  e.  ~P X  |  ( X  \  U ) 
C_  x } ) )  ->  ( U  i^i  ( X  \  U
) )  =/=  (/) )
1817ex 423 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  ->  ( A  e.  ( J  fClus  { x  e.  ~P X  |  ( X  \  U )  C_  x } )  ->  ( U  i^i  ( X  \  U ) )  =/=  (/) ) )
1918necon2bd 2508 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  ->  (
( U  i^i  ( X  \  U ) )  =  (/)  ->  -.  A  e.  ( J  fClus  { x  e.  ~P X  |  ( X  \  U ) 
C_  x } ) ) )
201, 19mpi 16 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  ->  -.  A  e.  ( J  fClus  { x  e.  ~P X  |  ( X  \  U )  C_  x } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   ` cfv 5271  (class class class)co 5874  TopOnctopon 16648    fClus cfcls 17647
This theorem is referenced by:  fclscf  17736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-top 16652  df-topon 16655  df-cld 16772  df-ntr 16773  df-cls 16774  df-fbas 17536  df-fil 17557  df-fcls 17652
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