Theorem 0ntop 22731

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 4323	. 2 ⊢ ¬ ∅ ∈ ∅
2		0opn 22730	. 2 ⊢ (∅ ∈ Top → ∅ ∈ ∅)
3	1, 2	mto 196	1 ⊢ ¬ ∅ ∈ Top

Syntax hints:  ¬ wn 3   ∈ wcel 2098  ∅c0 4315  Topctop 22719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2695  ax-sep 5290

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-in 3948  df-ss 3958  df-nul 4316  df-pw 4597  df-sn 4622  df-uni 4901  df-top 22720

This theorem is referenced by:  istps  22760  ordcmp  35823  onint1  35825  kelac1  42319