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Theorem ssntr 17153
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 3516 . . . . 5  |-  ( O  e.  ( J  i^i  ~P S )  <->  ( O  e.  J  /\  O  e. 
~P S ) )
2 elpwg 3830 . . . . . 6  |-  ( O  e.  J  ->  ( O  e.  ~P S  <->  O 
C_  S ) )
32pm5.32i 620 . . . . 5  |-  ( ( O  e.  J  /\  O  e.  ~P S
)  <->  ( O  e.  J  /\  O  C_  S ) )
41, 3bitr2i 243 . . . 4  |-  ( ( O  e.  J  /\  O  C_  S )  <->  O  e.  ( J  i^i  ~P S
) )
5 elssuni 4067 . . . 4  |-  ( O  e.  ( J  i^i  ~P S )  ->  O  C_ 
U. ( J  i^i  ~P S ) )
64, 5sylbi 189 . . 3  |-  ( ( O  e.  J  /\  O  C_  S )  ->  O  C_  U. ( J  i^i  ~P S ) )
76adantl 454 . 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 17131 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )
109adantr 453 . 2  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( O  e.  J  /\  O  C_  S ) )  ->  ( ( int `  J ) `  S )  =  U. ( J  i^i  ~P S
) )
117, 10sseqtr4d 3371 1  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( O  e.  J  /\  O  C_  S ) )  ->  O  C_  (
( int `  J
) `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727    i^i cin 3305    C_ wss 3306   ~Pcpw 3823   U.cuni 4039   ` cfv 5483   Topctop 16989   intcnt 17112
This theorem is referenced by:  ntrin  17156  neiint  17199  restntr  17277  cnntri  17366  xkococnlem  17722  iccntr  18883  bcthlem5  19312  ftc1  19957  lgamucov  24853  cvmlift2lem12  25032  cvmlift3lem7  25043  ftc1cnnc  26317  opnregcld  26371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-top 16994  df-ntr 17115
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