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Theorem sn0topon 16735
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 3762 . 2  |-  ~P (/)  =  { (/)
}
2 0ex 4150 . . 3  |-  (/)  e.  _V
3 distopon 16734 . . 3  |-  ( (/)  e.  _V  ->  ~P (/)  e.  (TopOn `  (/) ) )
42, 3ax-mp 8 . 2  |-  ~P (/)  e.  (TopOn `  (/) )
51, 4eqeltrri 2354 1  |-  { (/) }  e.  (TopOn `  (/) )
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ~Pcpw 3625   {csn 3640   ` cfv 5255  TopOnctopon 16632
This theorem is referenced by:  sn0top  16736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-top 16636  df-topon 16639
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