Theorem 0ntop 21507

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 4296	. 2 ⊢ ¬ ∅ ∈ ∅
2		0opn 21506	. 2 ⊢ (∅ ∈ Top → ∅ ∈ ∅)
3	1, 2	mto 199	1 ⊢ ¬ ∅ ∈ Top

Syntax hints:  ¬ wn 3   ∈ wcel 2110  ∅c0 4291  Topctop 21495

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-in 3943  df-ss 3952  df-nul 4292  df-pw 4541  df-sn 4562  df-uni 4833  df-top 21496

This theorem is referenced by:  istps  21536  ordcmp  33790  onint1  33792  kelac1  39656