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Theorem supnfcls 17715
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 3526 . 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 3303 . . . . . . 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 16666 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
8 elpw2g 4174 . . . . . . . 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 3197 . . . . . . 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 3200 . . . . . . 7  |-  ( x  =  ( X  \  U )  ->  (
( X  \  U
)  C_  x  <->  ( X  \  U )  C_  ( X  \  U ) ) )
1413elrab 2923 . . . . . 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 17710 . . . . 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 2495 . 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 1623    e. wcel 1684    =/= wne 2446   {crab 2547    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   ` cfv 5255  (class class class)co 5858  TopOnctopon 16632    fClus cfcls 17631
This theorem is referenced by:  fclscf  17720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-top 16636  df-topon 16639  df-cld 16756  df-ntr 16757  df-cls 16758  df-fbas 17520  df-fil 17541  df-fcls 17636
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