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Theorem ntrcls0 13984
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 109 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  J  e.  Top )
2 clscld.1 . . . . . 6  |-  X  = 
U. J
32clsss3 13983 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
42sscls 13973 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  C_  ( ( cls `  J ) `  S
) )
52ntrss 13972 . . . . 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 1248 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) ) )
763adant3 1018 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  (
( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `
 S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) ) )
8 sseq2 3191 . . . 4  |-  ( ( ( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/)  ->  ( ( ( int `  J ) `
 S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) )  <->  ( ( int `  J ) `  S )  C_  (/) ) )
983ad2ant3 1021 . . 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 147 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  (
( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `
 S )  C_  (/) )
11 ss0 3475 . 2  |-  ( ( ( int `  J
) `  S )  C_  (/)  ->  ( ( int `  J ) `  S
)  =  (/) )
1210, 11syl 14 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    /\ wa 104    <-> wb 105    /\ w3a 979    = wceq 1363    e. wcel 2158    C_ wss 3141   (/)c0 3434   U.cuni 3821   ` cfv 5228   Topctop 13850   intcnt 13946   clsccl 13947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-top 13851  df-cld 13948  df-ntr 13949  df-cls 13950
This theorem is referenced by: (None)
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