Theorem 0ntop 23235

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 4285	. 2 ⊢ ¬ ∅ ∈ ∅
2		0opn 23234	. 2 ⊢ (∅ ∈ Top → ∅ ∈ ∅)
3	1, 2	mto 200	1 ⊢ ¬ ∅ ∈ Top

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2146  ∅c0 4280  Topctop 23223

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-nul 4281  df-pw 4561  df-uni 4870  df-top 23224

This theorem is used by:  istps  23264  ordcmp  37362  onint1  37364  kelac1  44205