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Theorem nrmsep2 17100
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 2296 . . . . 5  |-  U. J  =  U. J
43cldopn 16784 . . . 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 16782 . . . . 5  |-  ( C  e.  ( Clsd `  J
)  ->  C  C_  U. J
)
9 reldisj 3511 . . . . 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 17099 . . 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 3548 . . . 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 2663 . 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 1632    e. wcel 1696   E.wrex 2557    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   U.cuni 3843   ` cfv 5271   Clsdccld 16769   clsccl 16771   Nrmcnrm 17054
This theorem is referenced by:  nrmsep  17101  isnrm2  17102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-top 16652  df-cld 16772  df-nrm 17061
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