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Theorem 0opn 22406
Description: The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
Assertion
Ref Expression
0opn (𝐽 ∈ Top → ∅ ∈ 𝐽)

Proof of Theorem 0opn
StepHypRef Expression
1 uni0 4940 . 2 ∅ = ∅
2 0ss 4397 . . 3 ∅ ⊆ 𝐽
3 uniopn 22399 . . 3 ((𝐽 ∈ Top ∧ ∅ ⊆ 𝐽) → ∅ ∈ 𝐽)
42, 3mpan2 690 . 2 (𝐽 ∈ Top → ∅ ∈ 𝐽)
51, 4eqeltrrid 2839 1 (𝐽 ∈ Top → ∅ ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wss 3949  c0 4323   cuni 4909  Topctop 22395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324  df-pw 4605  df-sn 4630  df-uni 4910  df-top 22396
This theorem is referenced by:  0ntop  22407  topgele  22432  tgclb  22473  0top  22486  en1top  22487  en2top  22488  topcld  22539  clsval2  22554  ntr0  22585  opnnei  22624  0nei  22632  restrcl  22661  rest0  22673  ordtrest2lem  22707  iocpnfordt  22719  icomnfordt  22720  cnindis  22796  isconn2  22918  kqtop  23249  mopn0  24007  locfinref  32821  ordtrest2NEWlem  32902  sxbrsigalem3  33271  cnambfre  36536
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