Theorem 0ntop 23062

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 23061	. 2 ⊢ (∅ ∈ Top → ∅ ∈ ∅)
3	1, 2	mto 200	1 ⊢ ¬ ∅ ∈ Top

Syntax hints:  ¬ wn 3   ∈ wcel 2143  ∅c0 4286  Topctop 23050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-in 3912  df-ss 3922  df-nul 4287  df-pw 4564  df-uni 4873  df-top 23051

This theorem is referenced by:  istps  23091  ordcmp  36958  onint1  36960  kelac1  43790