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Theorem ntrcls0 19795
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 457 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  J  e.  Top )
2 clscld.1 . . . . . 6  |-  X  = 
U. J
32clsss3 19778 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
42sscls 19775 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  C_  ( ( cls `  J ) `  S
) )
52ntrss 19774 . . . . 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 1228 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) ) )
763adant3 1016 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  (
( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `
 S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) ) )
8 sseq2 3521 . . . 4  |-  ( ( ( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/)  ->  ( ( ( int `  J ) `
 S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) )  <->  ( ( int `  J ) `  S )  C_  (/) ) )
983ad2ant3 1019 . . 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 210 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  (
( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `
 S )  C_  (/) )
11 ss0 3825 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    C_ wss 3471   (/)c0 3793   U.cuni 4251   ` cfv 5594   Topctop 19612   intcnt 19736   clsccl 19737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-top 19617  df-cld 19738  df-ntr 19739  df-cls 19740
This theorem is referenced by: (None)
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