Theorem 0ntop 22883

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 4279	. 2 ⊢ ¬ ∅ ∈ ∅
2		0opn 22882	. 2 ⊢ (∅ ∈ Top → ∅ ∈ ∅)
3	1, 2	mto 197	1 ⊢ ¬ ∅ ∈ Top

Syntax hints:  ¬ wn 3   ∈ wcel 2114  ∅c0 4274  Topctop 22871

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-in 3897  df-ss 3907  df-nul 4275  df-pw 4544  df-uni 4852  df-top 22872

This theorem is referenced by:  istps  22912  ordcmp  36648  onint1  36650  kelac1  43512