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Theorem nrmsep2 17084
Description: In a normal space, any two disjoint closed sets have the property that each one is a subset of an open set whose closure is disjoint from the other. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
nrmsep2  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  E. x  e.  J  ( C  C_  x  /\  ( ( ( cls `  J
) `  x )  i^i  D )  =  (/) ) )
Distinct variable groups:    x, C    x, D    x, J

Proof of Theorem nrmsep2
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  J  e.  Nrm )
2 simpr2 962 . . . 4  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  D  e.  ( Clsd `  J
) )
3 eqid 2283 . . . . 5  |-  U. J  =  U. J
43cldopn 16768 . . . 4  |-  ( D  e.  ( Clsd `  J
)  ->  ( U. J  \  D )  e.  J )
52, 4syl 15 . . 3  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  ( U. J  \  D )  e.  J )
6 simpr1 961 . . 3  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  C  e.  ( Clsd `  J
) )
7 simpr3 963 . . . 4  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  ( C  i^i  D )  =  (/) )
83cldss 16766 . . . . 5  |-  ( C  e.  ( Clsd `  J
)  ->  C  C_  U. J
)
9 reldisj 3498 . . . . 5  |-  ( C 
C_  U. J  ->  (
( C  i^i  D
)  =  (/)  <->  C  C_  ( U. J  \  D ) ) )
106, 8, 93syl 18 . . . 4  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  (
( C  i^i  D
)  =  (/)  <->  C  C_  ( U. J  \  D ) ) )
117, 10mpbid 201 . . 3  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  C  C_  ( U. J  \  D ) )
12 nrmsep3 17083 . . 3  |-  ( ( J  e.  Nrm  /\  ( ( U. J  \  D )  e.  J  /\  C  e.  ( Clsd `  J )  /\  C  C_  ( U. J  \  D ) ) )  ->  E. x  e.  J  ( C  C_  x  /\  ( ( cls `  J
) `  x )  C_  ( U. J  \  D ) ) )
131, 5, 6, 11, 12syl13anc 1184 . 2  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  E. x  e.  J  ( C  C_  x  /\  ( ( cls `  J ) `
 x )  C_  ( U. J  \  D
) ) )
14 ssdifin0 3535 . . . 4  |-  ( ( ( cls `  J
) `  x )  C_  ( U. J  \  D )  ->  (
( ( cls `  J
) `  x )  i^i  D )  =  (/) )
1514anim2i 552 . . 3  |-  ( ( C  C_  x  /\  ( ( cls `  J
) `  x )  C_  ( U. J  \  D ) )  -> 
( C  C_  x  /\  ( ( ( cls `  J ) `  x
)  i^i  D )  =  (/) ) )
1615reximi 2650 . 2  |-  ( E. x  e.  J  ( C  C_  x  /\  ( ( cls `  J
) `  x )  C_  ( U. J  \  D ) )  ->  E. x  e.  J  ( C  C_  x  /\  ( ( ( cls `  J ) `  x
)  i^i  D )  =  (/) ) )
1713, 16syl 15 1  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  E. x  e.  J  ( C  C_  x  /\  ( ( ( cls `  J
) `  x )  i^i  D )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   U.cuni 3827   ` cfv 5255   Clsdccld 16753   clsccl 16755   Nrmcnrm 17038
This theorem is referenced by:  nrmsep  17085  isnrm2  17086
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-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-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  df-nrm 17045
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