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Theorem cldss2 12301
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 12300 . . 3  |-  ( x  e.  ( Clsd `  J
)  ->  x  C_  X
)
3 velpw 3517 . . 3  |-  ( x  e.  ~P X  <->  x  C_  X
)
42, 3sylibr 133 . 2  |-  ( x  e.  ( Clsd `  J
)  ->  x  e.  ~P X )
54ssriv 3101 1  |-  ( Clsd `  J )  C_  ~P X
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480    C_ wss 3071   ~Pcpw 3510   U.cuni 3739   ` cfv 5126   Clsdccld 12287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134  ax-un 4358
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-opab 3993  df-mpt 3994  df-id 4218  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-iota 5091  df-fun 5128  df-fn 5129  df-fv 5134  df-top 12191  df-cld 12290
This theorem is referenced by: (None)
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