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Theorem topsn 17002
Description: The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4011). (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 17001 . . 3  |-  ( J  e.  (TopOn `  { A } )  ->  ( { (/) ,  { A } }  C_  J  /\  J  C_  ~P { A } ) )
21simprd 451 . 2  |-  ( J  e.  (TopOn `  { A } )  ->  J  C_ 
~P { A }
)
3 pwsn 4011 . . 3  |-  ~P { A }  =  { (/)
,  { A } }
41simpld 447 . . 3  |-  ( J  e.  (TopOn `  { A } )  ->  { (/) ,  { A } }  C_  J )
53, 4syl5eqss 3394 . 2  |-  ( J  e.  (TopOn `  { A } )  ->  ~P { A }  C_  J
)
62, 5eqssd 3367 1  |-  ( J  e.  (TopOn `  { A } )  ->  J  =  ~P { A }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {csn 3816   {cpr 3817   ` cfv 5456  TopOnctopon 16961
This theorem is referenced by:  restsn2  17237
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-top 16965  df-topon 16968
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