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Theorem topcld 19302
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 3895 . . . 4  |-  ( X 
\  X )  =  (/)
2 0opn 19180 . . . 4  |-  ( J  e.  Top  ->  (/)  e.  J
)
31, 2syl5eqel 2559 . . 3  |-  ( J  e.  Top  ->  ( X  \  X )  e.  J )
4 ssid 3523 . . 3  |-  X  C_  X
53, 4jctil 537 . 2  |-  ( J  e.  Top  ->  ( X  C_  X  /\  ( X  \  X )  e.  J ) )
6 iscld.1 . . 3  |-  X  = 
U. J
76iscld 19294 . 2  |-  ( J  e.  Top  ->  ( X  e.  ( Clsd `  J )  <->  ( X  C_  X  /\  ( X 
\  X )  e.  J ) ) )
85, 7mpbird 232 1  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3473    C_ wss 3476   (/)c0 3785   U.cuni 4245   ` cfv 5586   Topctop 19161   Clsdccld 19283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-top 19166  df-cld 19286
This theorem is referenced by:  clsval  19304  riincld  19311  clscld  19314  clstop  19336  cldmre  19345  indiscld  19358  iscon2  19681  cnmpt2pc  21163  rlmbn  21536  ubthlem1  25462  unicls  27521  cmpfiiin  30233  kelac1  30613
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