MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0ntop Structured version   Visualization version   GIF version

Theorem 0ntop 22277
Description: The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
Assertion
Ref Expression
0ntop ¬ ∅ ∈ Top

Proof of Theorem 0ntop
StepHypRef Expression
1 noel 4294 . 2 ¬ ∅ ∈ ∅
2 0opn 22276 . 2 (∅ ∈ Top → ∅ ∈ ∅)
31, 2mto 196 1 ¬ ∅ ∈ Top
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2107  c0 4286  Topctop 22265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5260
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-in 3921  df-ss 3931  df-nul 4287  df-pw 4566  df-sn 4591  df-uni 4870  df-top 22266
This theorem is referenced by:  istps  22306  ordcmp  34972  onint1  34974  kelac1  41437
  Copyright terms: Public domain W3C validator