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Theorem nrmsep2 20024
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 455 . . 3  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  J  e.  Nrm )
2 simpr2 1001 . . . 4  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  D  e.  ( Clsd `  J
) )
3 eqid 2454 . . . . 5  |-  U. J  =  U. J
43cldopn 19699 . . . 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 1000 . . 3  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  C  e.  ( Clsd `  J
) )
7 simpr3 1002 . . . 4  |-  ( ( J  e.  Nrm  /\  ( C  e.  ( Clsd `  J )  /\  D  e.  ( Clsd `  J )  /\  ( C  i^i  D )  =  (/) ) )  ->  ( C  i^i  D )  =  (/) )
83cldss 19697 . . . . 5  |-  ( C  e.  ( Clsd `  J
)  ->  C  C_  U. J
)
9 reldisj 3858 . . . . 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 20023 . . 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 1228 . 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 3897 . . . 4  |-  ( ( ( cls `  J
) `  x )  C_  ( U. J  \  D )  ->  (
( ( cls `  J
) `  x )  i^i  D )  =  (/) )
1514anim2i 567 . . 3  |-  ( ( C  C_  x  /\  ( ( cls `  J
) `  x )  C_  ( U. J  \  D ) )  -> 
( C  C_  x  /\  ( ( ( cls `  J ) `  x
)  i^i  D )  =  (/) ) )
1615reximi 2922 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E.wrex 2805    \ cdif 3458    i^i cin 3460    C_ wss 3461   (/)c0 3783   U.cuni 4235   ` cfv 5570   Clsdccld 19684   clsccl 19686   Nrmcnrm 19978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578  df-top 19566  df-cld 19687  df-nrm 19985
This theorem is referenced by:  nrmsep  20025  isnrm2  20026
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