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Theorem ssntr 20073
Description: An open subset of a set is a subset of the set's interior. (Contributed by Jeff Hankins, 31-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
ssntr  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( O  e.  J  /\  O  C_  S ) )  ->  O  C_  (
( int `  J
) `  S )
)

Proof of Theorem ssntr
StepHypRef Expression
1 elin 3617 . . . . 5  |-  ( O  e.  ( J  i^i  ~P S )  <->  ( O  e.  J  /\  O  e. 
~P S ) )
2 elpwg 3959 . . . . . 6  |-  ( O  e.  J  ->  ( O  e.  ~P S  <->  O 
C_  S ) )
32pm5.32i 643 . . . . 5  |-  ( ( O  e.  J  /\  O  e.  ~P S
)  <->  ( O  e.  J  /\  O  C_  S ) )
41, 3bitr2i 254 . . . 4  |-  ( ( O  e.  J  /\  O  C_  S )  <->  O  e.  ( J  i^i  ~P S
) )
5 elssuni 4227 . . . 4  |-  ( O  e.  ( J  i^i  ~P S )  ->  O  C_ 
U. ( J  i^i  ~P S ) )
64, 5sylbi 199 . . 3  |-  ( ( O  e.  J  /\  O  C_  S )  ->  O  C_  U. ( J  i^i  ~P S ) )
76adantl 468 . 2  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( O  e.  J  /\  O  C_  S ) )  ->  O  C_  U. ( J  i^i  ~P S ) )
8 clscld.1 . . . 4  |-  X  = 
U. J
98ntrval 20051 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )
109adantr 467 . 2  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( O  e.  J  /\  O  C_  S ) )  ->  ( ( int `  J ) `  S )  =  U. ( J  i^i  ~P S
) )
117, 10sseqtr4d 3469 1  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( O  e.  J  /\  O  C_  S ) )  ->  O  C_  (
( int `  J
) `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887    i^i cin 3403    C_ wss 3404   ~Pcpw 3951   U.cuni 4198   ` cfv 5582   Topctop 19917   intcnt 20032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-top 19921  df-ntr 20035
This theorem is referenced by:  ntrin  20076  neiint  20120  restntr  20198  cnntri  20287  xkococnlem  20674  iccntr  21839  bcthlem5  22296  ftc1  22994  lgamucov  23963  cvmlift2lem12  30037  cvmlift3lem7  30048  opnregcld  30986  ftc1cnnc  32016
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