MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  topontopi Unicode version

Theorem topontopi 16725
Description: A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypothesis
Ref Expression
topontopi.1  |-  J  e.  (TopOn `  B )
Assertion
Ref Expression
topontopi  |-  J  e. 
Top

Proof of Theorem topontopi
StepHypRef Expression
1 topontopi.1 . 2  |-  J  e.  (TopOn `  B )
2 topontop 16720 . 2  |-  ( J  e.  (TopOn `  B
)  ->  J  e.  Top )
31, 2ax-mp 8 1  |-  J  e. 
Top
Colors of variables: wff set class
Syntax hints:    e. wcel 1701   ` cfv 5292   Topctop 16687  TopOnctopon 16688
This theorem is referenced by:  sn0top  16792  indistop  16795  letop  16992  dfac14  17368  cnfldtop  18345  iitop  18436  limccnp2  19295  cxpcn3  20141  lmlim  23402  pnfneige0  23405  lmxrge0  23406  sxbrsigalem4  23811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-iota 5256  df-fun 5294  df-fv 5300  df-topon 16695
  Copyright terms: Public domain W3C validator