Theorem 0ntop 22406

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 4330	. 2 ⊢ ¬ ∅ ∈ ∅
2		0opn 22405	. 2 ⊢ (∅ ∈ Top → ∅ ∈ ∅)
3	1, 2	mto 196	1 ⊢ ¬ ∅ ∈ Top

Syntax hints:  ¬ wn 3   ∈ wcel 2106  ∅c0 4322  Topctop 22394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2703  ax-sep 5299

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323  df-pw 4604  df-sn 4629  df-uni 4909  df-top 22395

This theorem is referenced by:  istps  22435  ordcmp  35327  onint1  35329  kelac1  41795