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Theorem topsn 18667
Description: The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4188). (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 18666 . . 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 4188 . . 3  |-  ~P { A }  =  { (/)
,  { A } }
41simpld 459 . . 3  |-  ( J  e.  (TopOn `  { A } )  ->  { (/) ,  { A } }  C_  J )
53, 4syl5eqss 3503 . 2  |-  ( J  e.  (TopOn `  { A } )  ->  ~P { A }  C_  J
)
62, 5eqssd 3476 1  |-  ( J  e.  (TopOn `  { A } )  ->  J  =  ~P { A }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    C_ wss 3431   (/)c0 3740   ~Pcpw 3963   {csn 3980   {cpr 3982   ` cfv 5521  TopOnctopon 18626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-top 18630  df-topon 18633
This theorem is referenced by:  restsn2  18902
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