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Theorem ntrcls0 17140
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 444 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  J  e.  Top )
2 clscld.1 . . . . . 6  |-  X  = 
U. J
32clsss3 17123 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
42sscls 17120 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  C_  ( ( cls `  J ) `  S
) )
52ntrss 17119 . . . . 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 1184 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) ) )
763adant3 977 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  (
( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `
 S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) ) )
8 sseq2 3370 . . . 4  |-  ( ( ( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/)  ->  ( ( ( int `  J ) `
 S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) )  <->  ( ( int `  J ) `  S )  C_  (/) ) )
983ad2ant3 980 . . 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 202 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  (
( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `
 S )  C_  (/) )
11 ss0 3658 . 2  |-  ( ( ( int `  J
) `  S )  C_  (/)  ->  ( ( int `  J ) `  S
)  =  (/) )
1210, 11syl 16 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  (
( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `
 S )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3320   (/)c0 3628   U.cuni 4015   ` cfv 5454   Topctop 16958   intcnt 17081   clsccl 17082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-top 16963  df-cld 17083  df-ntr 17084  df-cls 17085
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