Theorem 0ntop 23030

Description: The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)

Assertion

Ref	Expression
0ntop	⊢ ¬ ∅ ∈ Top

Proof of Theorem 0ntop

Step	Hyp	Ref	Expression
1		noel 4299	. 2 ⊢ ¬ ∅ ∈ ∅
2		0opn 23029	. 2 ⊢ (∅ ∈ Top → ∅ ∈ ∅)
3	1, 2	mto 200	1 ⊢ ¬ ∅ ∈ Top

Syntax hints:  ¬ wn 3   ∈ wcel 2149  ∅c0 4294  Topctop 23018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-in 3920  df-ss 3930  df-nul 4295  df-pw 4569  df-uni 4877  df-top 23019

This theorem is referenced by:  istps  23059  ordcmp  36846  onint1  36848  kelac1  43681