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Theorem ntrcls0 20169
Description: A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
ntrcls0  |-  ( ( J  e.  Top  /\  S  C_  X  /\  (
( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `
 S )  =  (/) )

Proof of Theorem ntrcls0
StepHypRef Expression
1 simpl 464 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  J  e.  Top )
2 clscld.1 . . . . . 6  |-  X  = 
U. J
32clsss3 20151 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
42sscls 20148 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  C_  ( ( cls `  J ) `  S
) )
52ntrss 20147 . . . . 5  |-  ( ( J  e.  Top  /\  ( ( cls `  J
) `  S )  C_  X  /\  S  C_  ( ( cls `  J
) `  S )
)  ->  ( ( int `  J ) `  S )  C_  (
( int `  J
) `  ( ( cls `  J ) `  S ) ) )
61, 3, 4, 5syl3anc 1292 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) ) )
763adant3 1050 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  (
( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `
 S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) ) )
8 sseq2 3440 . . . 4  |-  ( ( ( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/)  ->  ( ( ( int `  J ) `
 S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) )  <->  ( ( int `  J ) `  S )  C_  (/) ) )
983ad2ant3 1053 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  (
( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( ( int `  J
) `  S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) )  <->  ( ( int `  J ) `  S )  C_  (/) ) )
107, 9mpbid 215 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  (
( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `
 S )  C_  (/) )
11 ss0 3768 . 2  |-  ( ( ( int `  J
) `  S )  C_  (/)  ->  ( ( int `  J ) `  S
)  =  (/) )
1210, 11syl 17 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  (
( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `
 S )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    C_ wss 3390   (/)c0 3722   U.cuni 4190   ` cfv 5589   Topctop 19994   intcnt 20109   clsccl 20110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-top 19998  df-cld 20111  df-ntr 20112  df-cls 20113
This theorem is referenced by: (None)
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