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Theorem cldss 14057
Description: A closed set is a subset of the underlying set of a topology. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)
Hypothesis
Ref Expression
iscld.1  |-  X  = 
U. J
Assertion
Ref Expression
cldss  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  X
)

Proof of Theorem cldss
StepHypRef Expression
1 cldrcl 14054 . 2  |-  ( S  e.  ( Clsd `  J
)  ->  J  e.  Top )
2 iscld.1 . . . 4  |-  X  = 
U. J
32iscld 14055 . . 3  |-  ( J  e.  Top  ->  ( S  e.  ( Clsd `  J )  <->  ( S  C_  X  /\  ( X 
\  S )  e.  J ) ) )
43simprbda 383 . 2  |-  ( ( J  e.  Top  /\  S  e.  ( Clsd `  J ) )  ->  S  C_  X )
51, 4mpancom 422 1  |-  ( S  e.  ( Clsd `  J
)  ->  S  C_  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160    \ cdif 3141    C_ wss 3144   U.cuni 3824   ` cfv 5235   Topctop 13949   Clsdccld 14044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-top 13950  df-cld 14047
This theorem is referenced by:  cldss2  14058  uncld  14065  cldcls  14066  clsss2  14081
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