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Theorem topsn 19203
Description: The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4239). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
topsn  |-  ( J  e.  (TopOn `  { A } )  ->  J  =  ~P { A }
)

Proof of Theorem topsn
StepHypRef Expression
1 topgele 19202 . . 3  |-  ( J  e.  (TopOn `  { A } )  ->  ( { (/) ,  { A } }  C_  J  /\  J  C_  ~P { A } ) )
21simprd 463 . 2  |-  ( J  e.  (TopOn `  { A } )  ->  J  C_ 
~P { A }
)
3 pwsn 4239 . . 3  |-  ~P { A }  =  { (/)
,  { A } }
41simpld 459 . . 3  |-  ( J  e.  (TopOn `  { A } )  ->  { (/) ,  { A } }  C_  J )
53, 4syl5eqss 3548 . 2  |-  ( J  e.  (TopOn `  { A } )  ->  ~P { A }  C_  J
)
62, 5eqssd 3521 1  |-  ( J  e.  (TopOn `  { A } )  ->  J  =  ~P { A }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   {cpr 4029   ` cfv 5586  TopOnctopon 19162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-top 19166  df-topon 19169
This theorem is referenced by:  restsn2  19438
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