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Theorem nrmsep2 17335
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 444 . . 3  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  J  e.  Nrm )
2 simpr2 964 . . . 4  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  D  e.  ( Clsd `  J
) )
3 eqid 2380 . . . . 5  |-  U. J  =  U. J
43cldopn 17011 . . . 4  |-  ( D  e.  ( Clsd `  J
)  ->  ( U. J  \  D )  e.  J )
52, 4syl 16 . . 3  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  ( U. J  \  D )  e.  J )
6 simpr1 963 . . 3  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  C  e.  ( Clsd `  J
) )
7 simpr3 965 . . . 4  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  ( C  i^i  D )  =  (/) )
83cldss 17009 . . . . 5  |-  ( C  e.  ( Clsd `  J
)  ->  C  C_  U. J
)
9 reldisj 3607 . . . . 5  |-  ( C 
C_  U. J  ->  (
( C  i^i  D
)  =  (/)  <->  C  C_  ( U. J  \  D ) ) )
106, 8, 93syl 19 . . . 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 202 . . 3  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  C  C_  ( U. J  \  D ) )
12 nrmsep3 17334 . . 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 1186 . 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 3645 . . . 4  |-  ( ( ( cls `  J
) `  x )  C_  ( U. J  \  D )  ->  (
( ( cls `  J
) `  x )  i^i  D )  =  (/) )
1514anim2i 553 . . 3  |-  ( ( C  C_  x  /\  ( ( cls `  J
) `  x )  C_  ( U. J  \  D ) )  -> 
( C  C_  x  /\  ( ( ( cls `  J ) `  x
)  i^i  D )  =  (/) ) )
1615reximi 2749 . 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 16 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   E.wrex 2643    \ cdif 3253    i^i cin 3255    C_ wss 3256   (/)c0 3564   U.cuni 3950   ` cfv 5387   Clsdccld 16996   clsccl 16998   Nrmcnrm 17289
This theorem is referenced by:  nrmsep  17336  isnrm2  17337
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 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fn 5390  df-fv 5395  df-top 16879  df-cld 16999  df-nrm 17296
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