Theorem 0ntop 23112

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 4291	. 2 ⊢ ¬ ∅ ∈ ∅
2		0opn 23111	. 2 ⊢ (∅ ∈ Top → ∅ ∈ ∅)
3	1, 2	mto 200	1 ⊢ ¬ ∅ ∈ Top

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2146  ∅c0 4286  Topctop 23100

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-in 3913  df-ss 3923  df-nul 4287  df-pw 4566  df-uni 4875  df-top 23101

This theorem is used by:  istps  23141  ordcmp  37015  onint1  37017  kelac1  43848