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Theorem intcld 18757
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 4319 . 2  |-  |^| A  =  |^|_ x  e.  A  x
2 dfss3 3441 . . 3  |-  ( A 
C_  ( Clsd `  J
)  <->  A. x  e.  A  x  e.  ( Clsd `  J ) )
3 iincld 18756 . . 3  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  x  e.  ( Clsd `  J
) )  ->  |^|_ x  e.  A  x  e.  ( Clsd `  J )
)
42, 3sylan2b 475 . 2  |-  ( ( A  =/=  (/)  /\  A  C_  ( Clsd `  J
) )  ->  |^|_ x  e.  A  x  e.  ( Clsd `  J )
)
51, 4syl5eqel 2541 1  |-  ( ( A  =/=  (/)  /\  A  C_  ( Clsd `  J
) )  ->  |^| A  e.  ( Clsd `  J
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758    =/= wne 2642   A.wral 2793    C_ wss 3423   (/)c0 3732   |^|cint 4223   |^|_ciin 4267   ` cfv 5513   Clsdccld 18733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-iin 4269  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-iota 5476  df-fun 5515  df-fn 5516  df-fv 5521  df-top 18616  df-cld 18736
This theorem is referenced by:  incld  18760  clscld  18764  cldmre  18795
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