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Theorem ssntr 19427
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 3692 . . . . 5  |-  ( O  e.  ( J  i^i  ~P S )  <->  ( O  e.  J  /\  O  e. 
~P S ) )
2 elpwg 4024 . . . . . 6  |-  ( O  e.  J  ->  ( O  e.  ~P S  <->  O 
C_  S ) )
32pm5.32i 637 . . . . 5  |-  ( ( O  e.  J  /\  O  e.  ~P S
)  <->  ( O  e.  J  /\  O  C_  S ) )
41, 3bitr2i 250 . . . 4  |-  ( ( O  e.  J  /\  O  C_  S )  <->  O  e.  ( J  i^i  ~P S
) )
5 elssuni 4281 . . . 4  |-  ( O  e.  ( J  i^i  ~P S )  ->  O  C_ 
U. ( J  i^i  ~P S ) )
64, 5sylbi 195 . . 3  |-  ( ( O  e.  J  /\  O  C_  S )  ->  O  C_  U. ( J  i^i  ~P S ) )
76adantl 466 . 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 19405 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )
109adantr 465 . 2  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( O  e.  J  /\  O  C_  S ) )  ->  ( ( int `  J ) `  S )  =  U. ( J  i^i  ~P S
) )
117, 10sseqtr4d 3546 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 369    = wceq 1379    e. wcel 1767    i^i cin 3480    C_ wss 3481   ~Pcpw 4016   U.cuni 4251   ` cfv 5594   Topctop 19263   intcnt 19386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 19268  df-ntr 19389
This theorem is referenced by:  ntrin  19430  neiint  19473  restntr  19551  cnntri  19640  xkococnlem  20028  iccntr  21194  bcthlem5  21635  ftc1  22311  lgamucov  28405  cvmlift2lem12  28584  cvmlift3lem7  28595  ftc1cnnc  30016  opnregcld  30075
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