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Theorem nrmsep2 19073
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 457 . . 3  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  J  e.  Nrm )
2 simpr2 995 . . . 4  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  D  e.  ( Clsd `  J
) )
3 eqid 2451 . . . . 5  |-  U. J  =  U. J
43cldopn 18748 . . . 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 994 . . 3  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  C  e.  ( Clsd `  J
) )
7 simpr3 996 . . . 4  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  ( C  i^i  D )  =  (/) )
83cldss 18746 . . . . 5  |-  ( C  e.  ( Clsd `  J
)  ->  C  C_  U. J
)
9 reldisj 3817 . . . . 5  |-  ( C 
C_  U. J  ->  (
( C  i^i  D
)  =  (/)  <->  C  C_  ( U. J  \  D ) ) )
106, 8, 93syl 20 . . . 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 210 . . 3  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  C  C_  ( U. J  \  D ) )
12 nrmsep3 19072 . . 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 1221 . 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 3855 . . . 4  |-  ( ( ( cls `  J
) `  x )  C_  ( U. J  \  D )  ->  (
( ( cls `  J
) `  x )  i^i  D )  =  (/) )
1514anim2i 569 . . 3  |-  ( ( C  C_  x  /\  ( ( cls `  J
) `  x )  C_  ( U. J  \  D ) )  -> 
( C  C_  x  /\  ( ( ( cls `  J ) `  x
)  i^i  D )  =  (/) ) )
1615reximi 2916 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2794    \ cdif 3420    i^i cin 3422    C_ wss 3423   (/)c0 3732   U.cuni 4186   ` cfv 5513   Clsdccld 18733   clsccl 18735   Nrmcnrm 19027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-iota 5476  df-fun 5515  df-fn 5516  df-fv 5521  df-top 18616  df-cld 18736  df-nrm 19034
This theorem is referenced by:  nrmsep  19074  isnrm2  19075
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