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Theorem ntrss 16792
Description: Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
ntrss  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( int `  J
) `  T )  C_  ( ( int `  J
) `  S )
)

Proof of Theorem ntrss
StepHypRef Expression
1 sscon 3310 . . . . . . 7  |-  ( T 
C_  S  ->  ( X  \  S )  C_  ( X  \  T ) )
21adantl 452 . . . . . 6  |-  ( ( S  C_  X  /\  T  C_  S )  -> 
( X  \  S
)  C_  ( X  \  T ) )
3 difss 3303 . . . . . 6  |-  ( X 
\  T )  C_  X
42, 3jctil 523 . . . . 5  |-  ( ( S  C_  X  /\  T  C_  S )  -> 
( ( X  \  T )  C_  X  /\  ( X  \  S
)  C_  ( X  \  T ) ) )
5 clscld.1 . . . . . . 7  |-  X  = 
U. J
65clsss 16791 . . . . . 6  |-  ( ( J  e.  Top  /\  ( X  \  T ) 
C_  X  /\  ( X  \  S )  C_  ( X  \  T ) )  ->  ( ( cls `  J ) `  ( X  \  S ) )  C_  ( ( cls `  J ) `  ( X  \  T ) ) )
763expb 1152 . . . . 5  |-  ( ( J  e.  Top  /\  ( ( X  \  T )  C_  X  /\  ( X  \  S
)  C_  ( X  \  T ) ) )  ->  ( ( cls `  J ) `  ( X  \  S ) ) 
C_  ( ( cls `  J ) `  ( X  \  T ) ) )
84, 7sylan2 460 . . . 4  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( ( cls `  J ) `  ( X  \  S ) ) 
C_  ( ( cls `  J ) `  ( X  \  T ) ) )
9 sscon 3310 . . . 4  |-  ( ( ( cls `  J
) `  ( X  \  S ) )  C_  ( ( cls `  J
) `  ( X  \  T ) )  -> 
( X  \  (
( cls `  J
) `  ( X  \  T ) ) ) 
C_  ( X  \ 
( ( cls `  J
) `  ( X  \  S ) ) ) )
108, 9syl 15 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( X  \ 
( ( cls `  J
) `  ( X  \  T ) ) ) 
C_  ( X  \ 
( ( cls `  J
) `  ( X  \  S ) ) ) )
11 sstr2 3186 . . . . 5  |-  ( T 
C_  S  ->  ( S  C_  X  ->  T  C_  X ) )
1211impcom 419 . . . 4  |-  ( ( S  C_  X  /\  T  C_  S )  ->  T  C_  X )
135ntrval2 16788 . . . 4  |-  ( ( J  e.  Top  /\  T  C_  X )  -> 
( ( int `  J
) `  T )  =  ( X  \ 
( ( cls `  J
) `  ( X  \  T ) ) ) )
1412, 13sylan2 460 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( ( int `  J ) `  T
)  =  ( X 
\  ( ( cls `  J ) `  ( X  \  T ) ) ) )
155ntrval2 16788 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  ( X  \ 
( ( cls `  J
) `  ( X  \  S ) ) ) )
1615adantrr 697 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( ( int `  J ) `  S
)  =  ( X 
\  ( ( cls `  J ) `  ( X  \  S ) ) ) )
1710, 14, 163sstr4d 3221 . 2  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( ( int `  J ) `  T
)  C_  ( ( int `  J ) `  S ) )
18173impb 1147 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( int `  J
) `  T )  C_  ( ( int `  J
) `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   U.cuni 3827   ` cfv 5255   Topctop 16631   intcnt 16754   clsccl 16755
This theorem is referenced by:  ntrin  16798  ntrcls0  16813  dvreslem  19259  dvres2lem  19260  dvaddbr  19287  dvmulbr  19288  dvcnvrelem2  19365  ntruni  26245  cldregopn  26249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-cld 16756  df-ntr 16757  df-cls 16758
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