Theorem 0ntop 22847

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 4288	. 2 ⊢ ¬ ∅ ∈ ∅
2		0opn 22846	. 2 ⊢ (∅ ∈ Top → ∅ ∈ ∅)
3	1, 2	mto 197	1 ⊢ ¬ ∅ ∈ Top

Syntax hints:  ¬ wn 3   ∈ wcel 2113  ∅c0 4283  Topctop 22835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-in 3906  df-ss 3916  df-nul 4284  df-pw 4554  df-uni 4862  df-top 22836

This theorem is referenced by:  istps  22876  ordcmp  36590  onint1  36592  kelac1  43247