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Theorem intcld 19992
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 4291 . 2  |-  |^| A  =  |^|_ x  e.  A  x
2 dfss3 3392 . . 3  |-  ( A 
C_  ( Clsd `  J
)  <->  A. x  e.  A  x  e.  ( Clsd `  J ) )
3 iincld 19991 . . 3  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  x  e.  ( Clsd `  J
) )  ->  |^|_ x  e.  A  x  e.  ( Clsd `  J )
)
42, 3sylan2b 477 . 2  |-  ( ( A  =/=  (/)  /\  A  C_  ( Clsd `  J
) )  ->  |^|_ x  e.  A  x  e.  ( Clsd `  J )
)
51, 4syl5eqel 2505 1  |-  ( ( A  =/=  (/)  /\  A  C_  ( Clsd `  J
) )  ->  |^| A  e.  ( Clsd `  J
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1872    =/= wne 2594   A.wral 2709    C_ wss 3374   (/)c0 3699   |^|cint 4193   |^|_ciin 4238   ` cfv 5539   Clsdccld 19968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-rab 2718  df-v 3019  df-sbc 3238  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4158  df-int 4194  df-iun 4239  df-iin 4240  df-br 4362  df-opab 4421  df-mpt 4422  df-id 4706  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-iota 5503  df-fun 5541  df-fn 5542  df-fv 5547  df-top 19858  df-cld 19971
This theorem is referenced by:  incld  19995  clscld  19999  cldmre  20031
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