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Theorem 0top 16721
Description: The singleton of the empty set is the only topology possible for an empty underlying set. (Contributed by NM, 9-Sep-2006.)
Assertion
Ref Expression
0top  |-  ( J  e.  Top  ->  ( U. J  =  (/)  <->  J  =  { (/) } ) )

Proof of Theorem 0top
StepHypRef Expression
1 olc 373 . . 3  |-  ( J  =  { (/) }  ->  ( J  =  (/)  \/  J  =  { (/) } ) )
2 0opn 16650 . . . . . 6  |-  ( J  e.  Top  ->  (/)  e.  J
)
3 n0i 3460 . . . . . 6  |-  ( (/)  e.  J  ->  -.  J  =  (/) )
42, 3syl 15 . . . . 5  |-  ( J  e.  Top  ->  -.  J  =  (/) )
54pm2.21d 98 . . . 4  |-  ( J  e.  Top  ->  ( J  =  (/)  ->  J  =  { (/) } ) )
6 idd 21 . . . 4  |-  ( J  e.  Top  ->  ( J  =  { (/) }  ->  J  =  { (/) } ) )
75, 6jaod 369 . . 3  |-  ( J  e.  Top  ->  (
( J  =  (/)  \/  J  =  { (/) } )  ->  J  =  { (/) } ) )
81, 7impbid2 195 . 2  |-  ( J  e.  Top  ->  ( J  =  { (/) }  <->  ( J  =  (/)  \/  J  =  { (/) } ) ) )
9 uni0b 3852 . . 3  |-  ( U. J  =  (/)  <->  J  C_  { (/) } )
10 sssn 3772 . . 3  |-  ( J 
C_  { (/) }  <->  ( J  =  (/)  \/  J  =  { (/) } ) )
119, 10bitr2i 241 . 2  |-  ( ( J  =  (/)  \/  J  =  { (/) } )  <->  U. J  =  (/) )
128, 11syl6rbb 253 1  |-  ( J  e.  Top  ->  ( U. J  =  (/)  <->  J  =  { (/) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    = wceq 1623    e. wcel 1684    C_ wss 3152   (/)c0 3455   {csn 3640   U.cuni 3827   Topctop 16631
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-uni 3828  df-top 16636
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