Theorem 0ntop 22160

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 4277	. 2 ⊢ ¬ ∅ ∈ ∅
2		0opn 22159	. 2 ⊢ (∅ ∈ Top → ∅ ∈ ∅)
3	1, 2	mto 196	1 ⊢ ¬ ∅ ∈ Top

Syntax hints:  ¬ wn 3   ∈ wcel 2105  ∅c0 4269  Topctop 22148

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-ext 2707  ax-sep 5243

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-in 3905  df-ss 3915  df-nul 4270  df-pw 4549  df-sn 4574  df-uni 4853  df-top 22149

This theorem is referenced by:  istps  22189  ordcmp  34732  onint1  34734  kelac1  41159