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Theorem intcld 19300
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 4372 . 2  |-  |^| A  =  |^|_ x  e.  A  x
2 dfss3 3487 . . 3  |-  ( A 
C_  ( Clsd `  J
)  <->  A. x  e.  A  x  e.  ( Clsd `  J ) )
3 iincld 19299 . . 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 2552 1  |-  ( ( A  =/=  (/)  /\  A  C_  ( Clsd `  J
) )  ->  |^| A  e.  ( Clsd `  J
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1762    =/= wne 2655   A.wral 2807    C_ wss 3469   (/)c0 3778   |^|cint 4275   |^|_ciin 4319   ` cfv 5579   Clsdccld 19276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fn 5582  df-fv 5587  df-top 19159  df-cld 19279
This theorem is referenced by:  incld  19303  clscld  19307  cldmre  19338
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