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Theorem sn0topon 12039
Description: The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
sn0topon  |-  { (/) }  e.  (TopOn `  (/) )

Proof of Theorem sn0topon
StepHypRef Expression
1 pw0 3614 . 2  |-  ~P (/)  =  { (/)
}
2 0ex 3995 . . 3  |-  (/)  e.  _V
3 distopon 12038 . . 3  |-  ( (/)  e.  _V  ->  ~P (/)  e.  (TopOn `  (/) ) )
42, 3ax-mp 7 . 2  |-  ~P (/)  e.  (TopOn `  (/) )
51, 4eqeltrri 2173 1  |-  { (/) }  e.  (TopOn `  (/) )
Colors of variables: wff set class
Syntax hints:    e. wcel 1448   _Vcvv 2641   (/)c0 3310   ~Pcpw 3457   {csn 3474   ` cfv 5059  TopOnctopon 11959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-top 11947  df-topon 11960
This theorem is referenced by:  sn0top  12040
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