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Theorem cldss2 17094
Description: The set of closed sets is contained in the powerset of the base. (Contributed by Mario Carneiro, 6-Jan-2014.)
Hypothesis
Ref Expression
iscld.1  |-  X  = 
U. J
Assertion
Ref Expression
cldss2  |-  ( Clsd `  J )  C_  ~P X

Proof of Theorem cldss2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 iscld.1 . . . 4  |-  X  = 
U. J
21cldss 17093 . . 3  |-  ( x  e.  ( Clsd `  J
)  ->  x  C_  X
)
3 vex 2959 . . . 4  |-  x  e. 
_V
43elpw 3805 . . 3  |-  ( x  e.  ~P X  <->  x  C_  X
)
52, 4sylibr 204 . 2  |-  ( x  e.  ( Clsd `  J
)  ->  x  e.  ~P X )
65ssriv 3352 1  |-  ( Clsd `  J )  C_  ~P X
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725    C_ wss 3320   ~Pcpw 3799   U.cuni 4015   ` cfv 5454   Clsdccld 17080
This theorem is referenced by:  cldmre  17142  cncls2  17337  fclscmp  18062  bcthlem5  19281  ubthlem1  22372  unicls  24301
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 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462  df-top 16963  df-cld 17083
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