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Theorem topcld 17023
Description: The underlying set of a topology is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 3-Oct-2006.)
Hypothesis
Ref Expression
iscld.1  |-  X  = 
U. J
Assertion
Ref Expression
topcld  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )

Proof of Theorem topcld
StepHypRef Expression
1 difid 3640 . . . 4  |-  ( X 
\  X )  =  (/)
2 0opn 16901 . . . 4  |-  ( J  e.  Top  ->  (/)  e.  J
)
31, 2syl5eqel 2472 . . 3  |-  ( J  e.  Top  ->  ( X  \  X )  e.  J )
4 ssid 3311 . . 3  |-  X  C_  X
53, 4jctil 524 . 2  |-  ( J  e.  Top  ->  ( X  C_  X  /\  ( X  \  X )  e.  J ) )
6 iscld.1 . . 3  |-  X  = 
U. J
76iscld 17015 . 2  |-  ( J  e.  Top  ->  ( X  e.  ( Clsd `  J )  <->  ( X  C_  X  /\  ( X 
\  X )  e.  J ) ) )
85, 7mpbird 224 1  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    \ cdif 3261    C_ wss 3264   (/)c0 3572   U.cuni 3958   ` cfv 5395   Topctop 16882   Clsdccld 17004
This theorem is referenced by:  clsval  17025  riincld  17032  clscld  17035  clstop  17057  cldmre  17066  indiscld  17079  iscon2  17399  cnmpt2pc  18825  rlmbn  19183  ubthlem1  22221  unicls  24106  cmpfiiin  26443  kelac1  26831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403  df-top 16887  df-cld 17007
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