Theorem 0ntop 22927

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 4344	. 2 ⊢ ¬ ∅ ∈ ∅
2		0opn 22926	. 2 ⊢ (∅ ∈ Top → ∅ ∈ ∅)
3	1, 2	mto 197	1 ⊢ ¬ ∅ ∈ Top

Syntax hints:  ¬ wn 3   ∈ wcel 2106  ∅c0 4339  Topctop 22915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-sn 4632  df-uni 4913  df-top 22916

This theorem is referenced by:  istps  22956  ordcmp  36430  onint1  36432  kelac1  43052