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

Theorem intcld 16877
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 4035 . 2  |-  |^| A  =  |^|_ x  e.  A  x
2 dfss3 3246 . . 3  |-  ( A 
C_  ( Clsd `  J
)  <->  A. x  e.  A  x  e.  ( Clsd `  J ) )
3 iincld 16876 . . 3  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  x  e.  ( Clsd `  J
) )  ->  |^|_ x  e.  A  x  e.  ( Clsd `  J )
)
42, 3sylan2b 461 . 2  |-  ( ( A  =/=  (/)  /\  A  C_  ( Clsd `  J
) )  ->  |^|_ x  e.  A  x  e.  ( Clsd `  J )
)
51, 4syl5eqel 2442 1  |-  ( ( A  =/=  (/)  /\  A  C_  ( Clsd `  J
) )  ->  |^| A  e.  ( Clsd `  J
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1710    =/= wne 2521   A.wral 2619    C_ wss 3228   (/)c0 3531   |^|cint 3941   |^|_ciin 3985   ` cfv 5334   Clsdccld 16853
This theorem is referenced by:  incld  16880  clscld  16884  cldmre  16915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-iota 5298  df-fun 5336  df-fn 5337  df-fv 5342  df-top 16736  df-cld 16856
  Copyright terms: Public domain W3C validator