MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  supnfcls Structured version   Unicode version

Theorem supnfcls 18054
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 3702 . 2  |-  ( U  i^i  ( X  \  U ) )  =  (/)
2 simpr 449 . . . . 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 962 . . . . 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 963 . . . . 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 3476 . . . . . . 7  |-  ( X 
\  U )  C_  X
6 simpl1 961 . . . . . . . 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 16995 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
8 elpw2g 4365 . . . . . . . 8  |-  ( X  e.  J  ->  (
( X  \  U
)  e.  ~P X  <->  ( X  \  U ) 
C_  X ) )
96, 7, 83syl 19 . . . . . . 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 226 . . . . . 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 3369 . . . . . . 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 3372 . . . . . . 7  |-  ( x  =  ( X  \  U )  ->  (
( X  \  U
)  C_  x  <->  ( X  \  U )  C_  ( X  \  U ) ) )
1413elrab 3094 . . . . . 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 647 . . . . 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 18049 . . . . 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 1187 . . . 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 425 . . 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 2655 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   {crab 2711    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   ` cfv 5456  (class class class)co 6083  TopOnctopon 16961    fClus cfcls 17970
This theorem is referenced by:  fclscf  18059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-fbas 16701  df-top 16965  df-topon 16968  df-cld 17085  df-ntr 17086  df-cls 17087  df-fil 17880  df-fcls 17975
  Copyright terms: Public domain W3C validator