Theorem 0ntop 22888

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 4266	. 2 ⊢ ¬ ∅ ∈ ∅
2		0opn 22887	. 2 ⊢ (∅ ∈ Top → ∅ ∈ ∅)
3	1, 2	mto 198	1 ⊢ ¬ ∅ ∈ Top

Syntax hints:  ¬ wn 3   ∈ wcel 2119  ∅c0 4261  Topctop 22876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4262  df-pw 4531  df-uni 4839  df-top 22877

This theorem is referenced by:  istps  22917  ordcmp  36675  onint1  36677  kelac1  43508