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Theorem 0ntr 16808
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 3513 . . . . 5  |-  ( X 
C_  S  <->  ( X  \  S )  =  (/) )
2 eqss 3194 . . . . . . . . 9  |-  ( S  =  X  <->  ( S  C_  X  /\  X  C_  S ) )
3 fveq2 5525 . . . . . . . . . . . . 13  |-  ( S  =  X  ->  (
( int `  J
) `  S )  =  ( ( int `  J ) `  X
) )
4 clscld.1 . . . . . . . . . . . . . 14  |-  X  = 
U. J
54ntrtop 16807 . . . . . . . . . . . . 13  |-  ( J  e.  Top  ->  (
( int `  J
) `  X )  =  X )
63, 5sylan9eqr 2337 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  S  =  X )  ->  ( ( int `  J
) `  S )  =  X )
76eqeq1d 2291 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  S  =  X )  ->  ( ( ( int `  J ) `  S
)  =  (/)  <->  X  =  (/) ) )
87biimpd 198 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  S  =  X )  ->  ( ( ( int `  J ) `  S
)  =  (/)  ->  X  =  (/) ) )
98ex 423 . . . . . . . . 9  |-  ( J  e.  Top  ->  ( S  =  X  ->  ( ( ( int `  J
) `  S )  =  (/)  ->  X  =  (/) ) ) )
102, 9syl5bir 209 . . . . . . . 8  |-  ( J  e.  Top  ->  (
( S  C_  X  /\  X  C_  S )  ->  ( ( ( int `  J ) `
 S )  =  (/)  ->  X  =  (/) ) ) )
1110exp3a 425 . . . . . . 7  |-  ( J  e.  Top  ->  ( S  C_  X  ->  ( X  C_  S  ->  (
( ( int `  J
) `  S )  =  (/)  ->  X  =  (/) ) ) ) )
1211com34 77 . . . . . 6  |-  ( J  e.  Top  ->  ( S  C_  X  ->  (
( ( int `  J
) `  S )  =  (/)  ->  ( X  C_  S  ->  X  =  (/) ) ) ) )
1312imp32 422 . . . . 5  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  -> 
( X  C_  S  ->  X  =  (/) ) )
141, 13syl5bir 209 . . . 4  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  -> 
( ( X  \  S )  =  (/)  ->  X  =  (/) ) )
1514necon3d 2484 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  -> 
( X  =/=  (/)  ->  ( X  \  S )  =/=  (/) ) )
1615imp 418 . 2  |-  ( ( ( J  e.  Top  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  /\  X  =/=  (/) )  ->  ( X  \  S )  =/=  (/) )
1716an32s 779 1  |-  ( ( ( J  e.  Top  /\  X  =/=  (/) )  /\  ( S  C_  X  /\  ( ( int `  J
) `  S )  =  (/) ) )  -> 
( X  \  S
)  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   (/)c0 3455   U.cuni 3827   ` cfv 5255   Topctop 16631   intcnt 16754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-ntr 16757
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