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Theorem distopon 17054
Description: The discrete topology on a set  A, with base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
distopon  |-  ( A  e.  V  ->  ~P A  e.  (TopOn `  A
) )

Proof of Theorem distopon
StepHypRef Expression
1 distop 17053 . 2  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 unipw 4407 . . . 4  |-  U. ~P A  =  A
32eqcomi 2440 . . 3  |-  A  = 
U. ~P A
43a1i 11 . 2  |-  ( A  e.  V  ->  A  =  U. ~P A )
5 istopon 16983 . 2  |-  ( ~P A  e.  (TopOn `  A )  <->  ( ~P A  e.  Top  /\  A  =  U. ~P A ) )
61, 4, 5sylanbrc 646 1  |-  ( A  e.  V  ->  ~P A  e.  (TopOn `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ~Pcpw 3792   U.cuni 4008   ` cfv 5447   Topctop 16951  TopOnctopon 16952
This theorem is referenced by:  sn0topon  17055  toponmre  17150  cndis  17348  txdis1cn  17660  xkofvcn  17709  distgp  18122  symgtgp  18124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fv 5455  df-top 16956  df-topon 16959
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