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Theorem ntrcls0 12082
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 108 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  J  e.  Top )
2 clscld.1 . . . . . 6  |-  X  = 
U. J
32clsss3 12081 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
42sscls 12071 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  C_  ( ( cls `  J ) `  S
) )
52ntrss 12070 . . . . 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 969 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  (
( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `
 S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) ) )
8 sseq2 3071 . . . 4  |-  ( ( ( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/)  ->  ( ( ( int `  J ) `
 S )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  S ) )  <->  ( ( int `  J ) `  S )  C_  (/) ) )
983ad2ant3 972 . . 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 146 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  (
( int `  J
) `  ( ( cls `  J ) `  S ) )  =  (/) )  ->  ( ( int `  J ) `
 S )  C_  (/) )
11 ss0 3350 . 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 103    <-> wb 104    /\ w3a 930    = wceq 1299    e. wcel 1448    C_ wss 3021   (/)c0 3310   U.cuni 3683   ` cfv 5059   Topctop 11946   intcnt 12044   clsccl 12045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-top 11947  df-cld 12046  df-ntr 12047  df-cls 12048
This theorem is referenced by: (None)
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