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Theorem 0top 16821
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 16750 . . . . . 6  |-  ( J  e.  Top  ->  (/)  e.  J
)
3 n0i 3536 . . . . . 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 3931 . . 3  |-  ( U. J  =  (/)  <->  J  C_  { (/) } )
10 sssn 3851 . . 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 1642    e. wcel 1710    C_ wss 3228   (/)c0 3531   {csn 3716   U.cuni 3906   Topctop 16731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-v 2866  df-dif 3231  df-in 3235  df-ss 3242  df-nul 3532  df-pw 3703  df-sn 3722  df-uni 3907  df-top 16736
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