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Theorem nrmsep3 17300
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 17280 . . . . . 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 3717 . . . . . . . 8  |-  ( y  =  A  ->  ~P y  =  ~P A
)
43ineq2d 3458 . . . . . . 7  |-  ( y  =  A  ->  (
( Clsd `  J )  i^i  ~P y )  =  ( ( Clsd `  J
)  i^i  ~P A
) )
5 sseq2 3286 . . . . . . . . 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 2649 . . . . . . 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 2833 . . . . . 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 2966 . . . . 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 3446 . . . . . 6  |-  ( B  e.  ( ( Clsd `  J )  i^i  ~P A )  <->  ( B  e.  ( Clsd `  J
)  /\  B  e.  ~P A ) )
12 elpwg 3721 . . . . . . 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 3285 . . . . . . . 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 2649 . . . . . 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 2966 . . . . 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 1167 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 935    = wceq 1647    e. wcel 1715   A.wral 2628   E.wrex 2629    i^i cin 3237    C_ wss 3238   ~Pcpw 3714   ` cfv 5358   Topctop 16848   Clsdccld 16970   clsccl 16972   Nrmcnrm 17255
This theorem is referenced by:  nrmsep2  17301  kqnrmlem1  17651  kqnrmlem2  17652  nrmr0reg  17657  nrmhmph  17702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-iota 5322  df-fv 5366  df-nrm 17262
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