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Theorem distopon 19261
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 19260 . 2  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 unipw 4697 . . . 4  |-  U. ~P A  =  A
32eqcomi 2480 . . 3  |-  A  = 
U. ~P A
43a1i 11 . 2  |-  ( A  e.  V  ->  A  =  U. ~P A )
5 istopon 19190 . 2  |-  ( ~P A  e.  (TopOn `  A )  <->  ( ~P A  e.  Top  /\  A  =  U. ~P A ) )
61, 4, 5sylanbrc 664 1  |-  ( A  e.  V  ->  ~P A  e.  (TopOn `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ~Pcpw 4010   U.cuni 4245   ` cfv 5586   Topctop 19158  TopOnctopon 19159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-top 19163  df-topon 19166
This theorem is referenced by:  sn0topon  19262  toponmre  19357  cndis  19555  txdis1cn  19868  xkofvcn  19917  distgp  20330  symgtgp  20332  cnfdmsn  31220
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