MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0top Structured version   Unicode version

Theorem 0top 17050
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 375 . . 3  |-  ( J  =  { (/) }  ->  ( J  =  (/)  \/  J  =  { (/) } ) )
2 0opn 16979 . . . . . 6  |-  ( J  e.  Top  ->  (/)  e.  J
)
3 n0i 3635 . . . . . 6  |-  ( (/)  e.  J  ->  -.  J  =  (/) )
42, 3syl 16 . . . . 5  |-  ( J  e.  Top  ->  -.  J  =  (/) )
54pm2.21d 101 . . . 4  |-  ( J  e.  Top  ->  ( J  =  (/)  ->  J  =  { (/) } ) )
6 idd 23 . . . 4  |-  ( J  e.  Top  ->  ( J  =  { (/) }  ->  J  =  { (/) } ) )
75, 6jaod 371 . . 3  |-  ( J  e.  Top  ->  (
( J  =  (/)  \/  J  =  { (/) } )  ->  J  =  { (/) } ) )
81, 7impbid2 197 . 2  |-  ( J  e.  Top  ->  ( J  =  { (/) }  <->  ( J  =  (/)  \/  J  =  { (/) } ) ) )
9 uni0b 4042 . . 3  |-  ( U. J  =  (/)  <->  J  C_  { (/) } )
10 sssn 3959 . . 3  |-  ( J 
C_  { (/) }  <->  ( J  =  (/)  \/  J  =  { (/) } ) )
119, 10bitr2i 243 . 2  |-  ( ( J  =  (/)  \/  J  =  { (/) } )  <->  U. J  =  (/) )
128, 11syl6rbb 255 1  |-  ( J  e.  Top  ->  ( U. J  =  (/)  <->  J  =  { (/) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    = wceq 1653    e. wcel 1726    C_ wss 3322   (/)c0 3630   {csn 3816   U.cuni 4017   Topctop 16960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-nul 3631  df-pw 3803  df-sn 3822  df-uni 4018  df-top 16965
  Copyright terms: Public domain W3C validator