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Theorem 0top 19612
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 384 . . 3  |-  ( J  =  { (/) }  ->  ( J  =  (/)  \/  J  =  { (/) } ) )
2 0opn 19540 . . . . . 6  |-  ( J  e.  Top  ->  (/)  e.  J
)
3 n0i 3798 . . . . . 6  |-  ( (/)  e.  J  ->  -.  J  =  (/) )
42, 3syl 16 . . . . 5  |-  ( J  e.  Top  ->  -.  J  =  (/) )
54pm2.21d 106 . . . 4  |-  ( J  e.  Top  ->  ( J  =  (/)  ->  J  =  { (/) } ) )
6 idd 24 . . . 4  |-  ( J  e.  Top  ->  ( J  =  { (/) }  ->  J  =  { (/) } ) )
75, 6jaod 380 . . 3  |-  ( J  e.  Top  ->  (
( J  =  (/)  \/  J  =  { (/) } )  ->  J  =  { (/) } ) )
81, 7impbid2 204 . 2  |-  ( J  e.  Top  ->  ( J  =  { (/) }  <->  ( J  =  (/)  \/  J  =  { (/) } ) ) )
9 uni0b 4276 . . 3  |-  ( U. J  =  (/)  <->  J  C_  { (/) } )
10 sssn 4190 . . 3  |-  ( J 
C_  { (/) }  <->  ( J  =  (/)  \/  J  =  { (/) } ) )
119, 10bitr2i 250 . 2  |-  ( ( J  =  (/)  \/  J  =  { (/) } )  <->  U. J  =  (/) )
128, 11syl6rbb 262 1  |-  ( J  e.  Top  ->  ( U. J  =  (/)  <->  J  =  { (/) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1395    e. wcel 1819    C_ wss 3471   (/)c0 3793   {csn 4032   U.cuni 4251   Topctop 19521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794  df-pw 4017  df-sn 4033  df-uni 4252  df-top 19526
This theorem is referenced by:  locfinref  28005
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