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Theorem 0ntr 17137
Description: A subset with an empty interior cannot cover a whole (nonempty) topology. (Contributed by NM, 12-Sep-2006.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
0ntr  |-  ( ( ( J  e.  Top  /\  X  =/=  (/) )  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  -> 
( X  \  S
)  =/=  (/) )

Proof of Theorem 0ntr
StepHypRef Expression
1 ssdif0 3688 . . . . 5  |-  ( X 
C_  S  <->  ( X  \  S )  =  (/) )
2 eqss 3365 . . . . . . . . 9  |-  ( S  =  X  <->  ( S  C_  X  /\  X  C_  S ) )
3 fveq2 5730 . . . . . . . . . . . . 13  |-  ( S  =  X  ->  (
( int `  J
) `  S )  =  ( ( int `  J ) `  X
) )
4 clscld.1 . . . . . . . . . . . . . 14  |-  X  = 
U. J
54ntrtop 17136 . . . . . . . . . . . . 13  |-  ( J  e.  Top  ->  (
( int `  J
) `  X )  =  X )
63, 5sylan9eqr 2492 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  S  =  X )  ->  ( ( int `  J
) `  S )  =  X )
76eqeq1d 2446 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  S  =  X )  ->  ( ( ( int `  J ) `  S
)  =  (/)  <->  X  =  (/) ) )
87biimpd 200 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  S  =  X )  ->  ( ( ( int `  J ) `  S
)  =  (/)  ->  X  =  (/) ) )
98ex 425 . . . . . . . . 9  |-  ( J  e.  Top  ->  ( S  =  X  ->  ( ( ( int `  J
) `  S )  =  (/)  ->  X  =  (/) ) ) )
102, 9syl5bir 211 . . . . . . . 8  |-  ( J  e.  Top  ->  (
( S  C_  X  /\  X  C_  S )  ->  ( ( ( int `  J ) `
 S )  =  (/)  ->  X  =  (/) ) ) )
1110exp3a 427 . . . . . . 7  |-  ( J  e.  Top  ->  ( S  C_  X  ->  ( X  C_  S  ->  (
( ( int `  J
) `  S )  =  (/)  ->  X  =  (/) ) ) ) )
1211com34 80 . . . . . 6  |-  ( J  e.  Top  ->  ( S  C_  X  ->  (
( ( int `  J
) `  S )  =  (/)  ->  ( X  C_  S  ->  X  =  (/) ) ) ) )
1312imp32 424 . . . . 5  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  -> 
( X  C_  S  ->  X  =  (/) ) )
141, 13syl5bir 211 . . . 4  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  -> 
( ( X  \  S )  =  (/)  ->  X  =  (/) ) )
1514necon3d 2641 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  -> 
( X  =/=  (/)  ->  ( X  \  S )  =/=  (/) ) )
1615imp 420 . 2  |-  ( ( ( J  e.  Top  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  /\  X  =/=  (/) )  ->  ( X  \  S )  =/=  (/) )
1716an32s 781 1  |-  ( ( ( J  e.  Top  /\  X  =/=  (/) )  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  -> 
( X  \  S
)  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601    \ cdif 3319    C_ wss 3322   (/)c0 3630   U.cuni 4017   ` cfv 5456   Topctop 16960   intcnt 17083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-top 16965  df-ntr 17086
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