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Theorem supnfcls 20284
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 3899 . 2  |-  ( U  i^i  ( X  \  U ) )  =  (/)
2 simpr 461 . . . . 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 1000 . . . . 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 1001 . . . . 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 3631 . . . . . . 7  |-  ( X 
\  U )  C_  X
6 simpl1 999 . . . . . . . 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 19224 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
8 elpw2g 4610 . . . . . . . 8  |-  ( X  e.  J  ->  (
( X  \  U
)  e.  ~P X  <->  ( X  \  U ) 
C_  X ) )
96, 7, 83syl 20 . . . . . . 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 233 . . . . . 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 3523 . . . . . . 7  |-  ( X 
\  U )  C_  ( X  \  U )
1211a1i 11 . . . . . 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 3526 . . . . . . 7  |-  ( x  =  ( X  \  U )  ->  (
( X  \  U
)  C_  x  <->  ( X  \  U )  C_  ( X  \  U ) ) )
1413elrab 3261 . . . . . 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 664 . . . . 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 20279 . . . . 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 1230 . . . 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 434 . . 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 2682 . 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 17 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   ` cfv 5588  (class class class)co 6284  TopOnctopon 19190    fClus cfcls 20200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-fbas 18215  df-top 19194  df-topon 19197  df-cld 19314  df-ntr 19315  df-cls 19316  df-fil 20110  df-fcls 20205
This theorem is referenced by:  fclscf  20289
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