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Theorem intcld 20132
Description: The intersection of a set of closed sets is closed. (Contributed by NM, 5-Oct-2006.)
Assertion
Ref Expression
intcld  |-  ( ( A  =/=  (/)  /\  A  C_  ( Clsd `  J
) )  ->  |^| A  e.  ( Clsd `  J
) )

Proof of Theorem intcld
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 intiin 4323 . 2  |-  |^| A  =  |^|_ x  e.  A  x
2 dfss3 3408 . . 3  |-  ( A 
C_  ( Clsd `  J
)  <->  A. x  e.  A  x  e.  ( Clsd `  J ) )
3 iincld 20131 . . 3  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  x  e.  ( Clsd `  J
) )  ->  |^|_ x  e.  A  x  e.  ( Clsd `  J )
)
42, 3sylan2b 483 . 2  |-  ( ( A  =/=  (/)  /\  A  C_  ( Clsd `  J
) )  ->  |^|_ x  e.  A  x  e.  ( Clsd `  J )
)
51, 4syl5eqel 2553 1  |-  ( ( A  =/=  (/)  /\  A  C_  ( Clsd `  J
) )  ->  |^| A  e.  ( Clsd `  J
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    e. wcel 1904    =/= wne 2641   A.wral 2756    C_ wss 3390   (/)c0 3722   |^|cint 4226   |^|_ciin 4270   ` cfv 5589   Clsdccld 20108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597  df-top 19998  df-cld 20111
This theorem is referenced by:  incld  20135  clscld  20139  cldmre  20171
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