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Theorem topsn 19726
Description: The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4184). (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 19725 . . 3  |-  ( J  e.  (TopOn `  { A } )  ->  ( { (/) ,  { A } }  C_  J  /\  J  C_  ~P { A } ) )
21simprd 461 . 2  |-  ( J  e.  (TopOn `  { A } )  ->  J  C_ 
~P { A }
)
3 pwsn 4184 . . 3  |-  ~P { A }  =  { (/)
,  { A } }
41simpld 457 . . 3  |-  ( J  e.  (TopOn `  { A } )  ->  { (/) ,  { A } }  C_  J )
53, 4syl5eqss 3485 . 2  |-  ( J  e.  (TopOn `  { A } )  ->  ~P { A }  C_  J
)
62, 5eqssd 3458 1  |-  ( J  e.  (TopOn `  { A } )  ->  J  =  ~P { A }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    C_ wss 3413   (/)c0 3737   ~Pcpw 3954   {csn 3971   {cpr 3973   ` cfv 5568  TopOnctopon 19685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-top 19689  df-topon 19692
This theorem is referenced by:  restsn2  19963
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