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Theorem ssntr 20150
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 3608 . . . . 5  |-  ( O  e.  ( J  i^i  ~P S )  <->  ( O  e.  J  /\  O  e. 
~P S ) )
2 elpwg 3950 . . . . . 6  |-  ( O  e.  J  ->  ( O  e.  ~P S  <->  O 
C_  S ) )
32pm5.32i 649 . . . . 5  |-  ( ( O  e.  J  /\  O  e.  ~P S
)  <->  ( O  e.  J  /\  O  C_  S ) )
41, 3bitr2i 258 . . . 4  |-  ( ( O  e.  J  /\  O  C_  S )  <->  O  e.  ( J  i^i  ~P S
) )
5 elssuni 4219 . . . 4  |-  ( O  e.  ( J  i^i  ~P S )  ->  O  C_ 
U. ( J  i^i  ~P S ) )
64, 5sylbi 200 . . 3  |-  ( ( O  e.  J  /\  O  C_  S )  ->  O  C_  U. ( J  i^i  ~P S ) )
76adantl 473 . 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 20128 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )
109adantr 472 . 2  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( O  e.  J  /\  O  C_  S ) )  ->  ( ( int `  J ) `  S )  =  U. ( J  i^i  ~P S
) )
117, 10sseqtr4d 3455 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 376    = wceq 1452    e. wcel 1904    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   U.cuni 4190   ` cfv 5589   Topctop 19994   intcnt 20109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-top 19998  df-ntr 20112
This theorem is referenced by:  ntrin  20153  neiint  20197  restntr  20275  cnntri  20364  xkococnlem  20751  iccntr  21917  bcthlem5  22374  ftc1  23073  lgamucov  24042  cvmlift2lem12  30109  cvmlift3lem7  30120  opnregcld  31057  ftc1cnnc  32080
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