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Theorem topcld 17062
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 3664 . . . 4  |-  ( X 
\  X )  =  (/)
2 0opn 16940 . . . 4  |-  ( J  e.  Top  ->  (/)  e.  J
)
31, 2syl5eqel 2496 . . 3  |-  ( J  e.  Top  ->  ( X  \  X )  e.  J )
4 ssid 3335 . . 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 17054 . 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 1721    \ cdif 3285    C_ wss 3288   (/)c0 3596   U.cuni 3983   ` cfv 5421   Topctop 16921   Clsdccld 17043
This theorem is referenced by:  clsval  17064  riincld  17071  clscld  17074  clstop  17096  cldmre  17105  indiscld  17118  iscon2  17438  cnmpt2pc  18914  rlmbn  19276  ubthlem1  22333  unicls  24262  cmpfiiin  26649  kelac1  27037
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-top 16926  df-cld 17046
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