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Theorem ntrss3 16853
Description: The interior of a subset of a topological space is included in the space. (Contributed by NM, 1-Oct-2007.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
ntrss3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  C_  X )

Proof of Theorem ntrss3
StepHypRef Expression
1 clscld.1 . . 3  |-  X  = 
U. J
21ntropn 16842 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  e.  J )
31eltopss 16709 . 2  |-  ( ( J  e.  Top  /\  ( ( int `  J
) `  S )  e.  J )  ->  (
( int `  J
) `  S )  C_  X )
42, 3syldan 456 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701    C_ wss 3186   U.cuni 3864   ` cfv 5292   Topctop 16687   intcnt 16810
This theorem is referenced by:  ntridm  16861  hmeontr  17516  bcthlem5  18803  perfdvf  19306  ubthlem1  21504  opnregcld  25397  cldregopn  25398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-top 16692  df-ntr 16813
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