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Theorem 0top 17038
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 374 . . 3  |-  ( J  =  { (/) }  ->  ( J  =  (/)  \/  J  =  { (/) } ) )
2 0opn 16967 . . . . . 6  |-  ( J  e.  Top  ->  (/)  e.  J
)
3 n0i 3625 . . . . . 6  |-  ( (/)  e.  J  ->  -.  J  =  (/) )
42, 3syl 16 . . . . 5  |-  ( J  e.  Top  ->  -.  J  =  (/) )
54pm2.21d 100 . . . 4  |-  ( J  e.  Top  ->  ( J  =  (/)  ->  J  =  { (/) } ) )
6 idd 22 . . . 4  |-  ( J  e.  Top  ->  ( J  =  { (/) }  ->  J  =  { (/) } ) )
75, 6jaod 370 . . 3  |-  ( J  e.  Top  ->  (
( J  =  (/)  \/  J  =  { (/) } )  ->  J  =  { (/) } ) )
81, 7impbid2 196 . 2  |-  ( J  e.  Top  ->  ( J  =  { (/) }  <->  ( J  =  (/)  \/  J  =  { (/) } ) ) )
9 uni0b 4032 . . 3  |-  ( U. J  =  (/)  <->  J  C_  { (/) } )
10 sssn 3949 . . 3  |-  ( J 
C_  { (/) }  <->  ( J  =  (/)  \/  J  =  { (/) } ) )
119, 10bitr2i 242 . 2  |-  ( ( J  =  (/)  \/  J  =  { (/) } )  <->  U. J  =  (/) )
128, 11syl6rbb 254 1  |-  ( J  e.  Top  ->  ( U. J  =  (/)  <->  J  =  { (/) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    = wceq 1652    e. wcel 1725    C_ wss 3312   (/)c0 3620   {csn 3806   U.cuni 4007   Topctop 16948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-pw 3793  df-sn 3812  df-uni 4008  df-top 16953
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