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Theorem nrmsep3 17083
Description: In a normal space, given a closed set  B inside an open set  A, there is an open set  x such that  B  C_  x  C_  cls ( x )  C_  A. (Contributed by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
nrmsep3  |-  ( ( J  e.  Nrm  /\  ( A  e.  J  /\  B  e.  ( Clsd `  J )  /\  B  C_  A ) )  ->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J
) `  x )  C_  A ) )
Distinct variable groups:    x, A    x, B    x, J

Proof of Theorem nrmsep3
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnrm 17063 . . . . . 6  |-  ( J  e.  Nrm  <->  ( J  e.  Top  /\  A. y  e.  J  A. z  e.  ( ( Clsd `  J
)  i^i  ~P y
) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J ) `  x
)  C_  y )
) )
21simprbi 450 . . . . 5  |-  ( J  e.  Nrm  ->  A. y  e.  J  A. z  e.  ( ( Clsd `  J
)  i^i  ~P y
) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J ) `  x
)  C_  y )
)
3 pweq 3628 . . . . . . . 8  |-  ( y  =  A  ->  ~P y  =  ~P A
)
43ineq2d 3370 . . . . . . 7  |-  ( y  =  A  ->  (
( Clsd `  J )  i^i  ~P y )  =  ( ( Clsd `  J
)  i^i  ~P A
) )
5 sseq2 3200 . . . . . . . . 9  |-  ( y  =  A  ->  (
( ( cls `  J
) `  x )  C_  y  <->  ( ( cls `  J ) `  x
)  C_  A )
)
65anbi2d 684 . . . . . . . 8  |-  ( y  =  A  ->  (
( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  y )  <->  ( z  C_  x  /\  ( ( cls `  J ) `
 x )  C_  A ) ) )
76rexbidv 2564 . . . . . . 7  |-  ( y  =  A  ->  ( E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  y )  <->  E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J ) `
 x )  C_  A ) ) )
84, 7raleqbidv 2748 . . . . . 6  |-  ( y  =  A  ->  ( A. z  e.  (
( Clsd `  J )  i^i  ~P y ) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  y )  <->  A. z  e.  ( ( Clsd `  J
)  i^i  ~P A
) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J ) `  x
)  C_  A )
) )
98rspccv 2881 . . . . 5  |-  ( A. y  e.  J  A. z  e.  ( ( Clsd `  J )  i^i 
~P y ) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  y )  ->  ( A  e.  J  ->  A. z  e.  ( (
Clsd `  J )  i^i  ~P A ) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  A ) ) )
102, 9syl 15 . . . 4  |-  ( J  e.  Nrm  ->  ( A  e.  J  ->  A. z  e.  ( (
Clsd `  J )  i^i  ~P A ) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  A ) ) )
11 elin 3358 . . . . . 6  |-  ( B  e.  ( ( Clsd `  J )  i^i  ~P A )  <->  ( B  e.  ( Clsd `  J
)  /\  B  e.  ~P A ) )
12 elpwg 3632 . . . . . . 7  |-  ( B  e.  ( Clsd `  J
)  ->  ( B  e.  ~P A  <->  B  C_  A
) )
1312pm5.32i 618 . . . . . 6  |-  ( ( B  e.  ( Clsd `  J )  /\  B  e.  ~P A )  <->  ( B  e.  ( Clsd `  J
)  /\  B  C_  A
) )
1411, 13bitri 240 . . . . 5  |-  ( B  e.  ( ( Clsd `  J )  i^i  ~P A )  <->  ( B  e.  ( Clsd `  J
)  /\  B  C_  A
) )
15 sseq1 3199 . . . . . . . 8  |-  ( z  =  B  ->  (
z  C_  x  <->  B  C_  x
) )
1615anbi1d 685 . . . . . . 7  |-  ( z  =  B  ->  (
( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  A )  <->  ( B  C_  x  /\  ( ( cls `  J ) `
 x )  C_  A ) ) )
1716rexbidv 2564 . . . . . 6  |-  ( z  =  B  ->  ( E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  A )  <->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J ) `
 x )  C_  A ) ) )
1817rspccv 2881 . . . . 5  |-  ( A. z  e.  ( ( Clsd `  J )  i^i 
~P A ) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  A )  ->  ( B  e.  ( ( Clsd `  J )  i^i 
~P A )  ->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J
) `  x )  C_  A ) ) )
1914, 18syl5bir 209 . . . 4  |-  ( A. z  e.  ( ( Clsd `  J )  i^i 
~P A ) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  A )  ->  (
( B  e.  (
Clsd `  J )  /\  B  C_  A )  ->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J
) `  x )  C_  A ) ) )
2010, 19syl6 29 . . 3  |-  ( J  e.  Nrm  ->  ( A  e.  J  ->  ( ( B  e.  (
Clsd `  J )  /\  B  C_  A )  ->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J
) `  x )  C_  A ) ) ) )
2120exp4a 589 . 2  |-  ( J  e.  Nrm  ->  ( A  e.  J  ->  ( B  e.  ( Clsd `  J )  ->  ( B  C_  A  ->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J ) `
 x )  C_  A ) ) ) ) )
22213imp2 1166 1  |-  ( ( J  e.  Nrm  /\  ( A  e.  J  /\  B  e.  ( Clsd `  J )  /\  B  C_  A ) )  ->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J
) `  x )  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   ` cfv 5255   Topctop 16631   Clsdccld 16753   clsccl 16755   Nrmcnrm 17038
This theorem is referenced by:  nrmsep2  17084  kqnrmlem1  17434  kqnrmlem2  17435  nrmr0reg  17440  nrmhmph  17485
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  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-iota 5219  df-fv 5263  df-nrm 17045
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