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Theorem cldmre 16831
Description: The closed sets of a topology comprise a Moore system on the points of the topology. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
cldmre  |-  ( J  e.  Top  ->  ( Clsd `  J )  e.  (Moore `  X )
)

Proof of Theorem cldmre
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 clscld.1 . . . 4  |-  X  = 
U. J
21cldss2 16783 . . 3  |-  ( Clsd `  J )  C_  ~P X
32a1i 10 . 2  |-  ( J  e.  Top  ->  ( Clsd `  J )  C_  ~P X )
41topcld 16788 . 2  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
5 intcld 16793 . . . 4  |-  ( ( x  =/=  (/)  /\  x  C_  ( Clsd `  J
) )  ->  |^| x  e.  ( Clsd `  J
) )
65ancoms 439 . . 3  |-  ( ( x  C_  ( Clsd `  J )  /\  x  =/=  (/) )  ->  |^| x  e.  ( Clsd `  J
) )
763adant1 973 . 2  |-  ( ( J  e.  Top  /\  x  C_  ( Clsd `  J
)  /\  x  =/=  (/) )  ->  |^| x  e.  ( Clsd `  J
) )
83, 4, 7ismred 13520 1  |-  ( J  e.  Top  ->  ( Clsd `  J )  e.  (Moore `  X )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   U.cuni 3843   |^|cint 3878   ` cfv 5271  Moorecmre 13500   Topctop 16647   Clsdccld 16769
This theorem is referenced by:  mrccls  16832  cldmreon  16847  mreclatdemo  16849
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-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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  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-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-mre 13504  df-top 16652  df-cld 16772
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