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Theorem cldmre 17135
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 17087 . . 3  |-  ( Clsd `  J )  C_  ~P X
32a1i 11 . 2  |-  ( J  e.  Top  ->  ( Clsd `  J )  C_  ~P X )
41topcld 17092 . 2  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
5 intcld 17097 . . . 4  |-  ( ( x  =/=  (/)  /\  x  C_  ( Clsd `  J
) )  ->  |^| x  e.  ( Clsd `  J
) )
65ancoms 440 . . 3  |-  ( ( x  C_  ( Clsd `  J )  /\  x  =/=  (/) )  ->  |^| x  e.  ( Clsd `  J
) )
763adant1 975 . 2  |-  ( ( J  e.  Top  /\  x  C_  ( Clsd `  J
)  /\  x  =/=  (/) )  ->  |^| x  e.  ( Clsd `  J
) )
83, 4, 7ismred 13820 1  |-  ( J  e.  Top  ->  ( Clsd `  J )  e.  (Moore `  X )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2599    C_ wss 3313   (/)c0 3621   ~Pcpw 3792   U.cuni 4008   |^|cint 4043   ` cfv 5447  Moorecmre 13800   Topctop 16951   Clsdccld 17073
This theorem is referenced by:  mrccls  17136  cldmreon  17151  mreclatdemo  17153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-iin 4089  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fn 5450  df-fv 5455  df-mre 13804  df-top 16956  df-cld 17076
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