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Theorem supnfcls 20606
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 3816 . 2  |-  ( U  i^i  ( X  \  U ) )  =  (/)
2 simpr 459 . . . . 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 998 . . . . 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 999 . . . . 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 3545 . . . . . . 7  |-  ( X 
\  U )  C_  X
6 simpl1 997 . . . . . . . 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 19514 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
8 elpw2g 4528 . . . . . . . 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 3436 . . . . . . 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 3439 . . . . . . 7  |-  ( x  =  ( X  \  U )  ->  (
( X  \  U
)  C_  x  <->  ( X  \  U )  C_  ( X  \  U ) ) )
1413elrab 3182 . . . . . 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 662 . . . . 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 20601 . . . . 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 1228 . . . 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 432 . . 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 2597 . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   {crab 2736    \ cdif 3386    i^i cin 3388    C_ wss 3389   (/)c0 3711   ~Pcpw 3927   ` cfv 5496  (class class class)co 6196  TopOnctopon 19480    fClus cfcls 20522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-fbas 18529  df-top 19484  df-topon 19487  df-cld 19605  df-ntr 19606  df-cls 19607  df-fil 20432  df-fcls 20527
This theorem is referenced by:  fclscf  20611
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