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Theorem incld 17030
Description: The intersection of two closed sets is closed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
incld  |-  ( ( A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  J
) )  ->  ( A  i^i  B )  e.  ( Clsd `  J
) )

Proof of Theorem incld
StepHypRef Expression
1 intprg 4026 . 2  |-  ( ( A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  J
) )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
2 prnzg 3867 . . . 4  |-  ( A  e.  ( Clsd `  J
)  ->  { A ,  B }  =/=  (/) )
32adantr 452 . . 3  |-  ( ( A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  J
) )  ->  { A ,  B }  =/=  (/) )
4 prssi 3897 . . 3  |-  ( ( A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  J
) )  ->  { A ,  B }  C_  ( Clsd `  J ) )
5 intcld 17027 . . 3  |-  ( ( { A ,  B }  =/=  (/)  /\  { A ,  B }  C_  ( Clsd `  J ) )  ->  |^| { A ,  B }  e.  ( Clsd `  J ) )
63, 4, 5syl2anc 643 . 2  |-  ( ( A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  J
) )  ->  |^| { A ,  B }  e.  (
Clsd `  J )
)
71, 6eqeltrrd 2462 1  |-  ( ( A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  J
) )  ->  ( A  i^i  B )  e.  ( Clsd `  J
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717    =/= wne 2550    i^i cin 3262    C_ wss 3263   (/)c0 3571   {cpr 3758   |^|cint 3992   ` cfv 5394   Clsdccld 17003
This theorem is referenced by:  riincld  17031  restcldr  17160  ordtcld3  17185  clsocv  19075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fn 5397  df-fv 5402  df-top 16886  df-cld 17006
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