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Theorem distopon 12167
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 12165 . 2  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 unipw 4109 . . . 4  |-  U. ~P A  =  A
32eqcomi 2121 . . 3  |-  A  = 
U. ~P A
43a1i 9 . 2  |-  ( A  e.  V  ->  A  =  U. ~P A )
5 istopon 12091 . 2  |-  ( ~P A  e.  (TopOn `  A )  <->  ( ~P A  e.  Top  /\  A  =  U. ~P A ) )
61, 4, 5sylanbrc 413 1  |-  ( A  e.  V  ->  ~P A  e.  (TopOn `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   ~Pcpw 3480   U.cuni 3706   ` cfv 5093   Topctop 12075  TopOnctopon 12088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-top 12076  df-topon 12089
This theorem is referenced by:  sn0topon  12168  cndis  12321  txdis1cn  12358
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