MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  incld Unicode version

Theorem incld 16780
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 3896 . 2  |-  ( ( A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  J
) )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
2 prnzg 3746 . . . 4  |-  ( A  e.  ( Clsd `  J
)  ->  { A ,  B }  =/=  (/) )
32adantr 451 . . 3  |-  ( ( A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  J
) )  ->  { A ,  B }  =/=  (/) )
4 prssi 3771 . . 3  |-  ( ( A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  J
) )  ->  { A ,  B }  C_  ( Clsd `  J ) )
5 intcld 16777 . . 3  |-  ( ( { A ,  B }  =/=  (/)  /\  { A ,  B }  C_  ( Clsd `  J ) )  ->  |^| { A ,  B }  e.  ( Clsd `  J ) )
63, 4, 5syl2anc 642 . 2  |-  ( ( A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  J
) )  ->  |^| { A ,  B }  e.  (
Clsd `  J )
)
71, 6eqeltrrd 2358 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 358    e. wcel 1684    =/= wne 2446    i^i cin 3151    C_ wss 3152   (/)c0 3455   {cpr 3641   |^|cint 3862   ` cfv 5255   Clsdccld 16753
This theorem is referenced by:  riincld  16781  restcldr  16905  ordtcld3  16929  clsocv  18677  incldOLD  26465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-top 16636  df-cld 16756
  Copyright terms: Public domain W3C validator