Theorem 0opn 21961

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

Step	Hyp	Ref	Expression
1		uni0 4866	. 2 ⊢ ∪ ∅ = ∅
2		0ss 4327	. . 3 ⊢ ∅ ⊆ 𝐽
3		uniopn 21954	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ 𝐽) → ∪ ∅ ∈ 𝐽)
4	2, 3	mpan2 687	. 2 ⊢ (𝐽 ∈ Top → ∪ ∅ ∈ 𝐽)
5	1, 4	eqeltrrid 2844	1 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)

Syntax hints:   → wi 4   ∈ wcel 2108   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836  Topctop 21950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-pw 4532  df-sn 4559  df-uni 4837  df-top 21951

This theorem is referenced by:  0ntop  21962  topgele  21987  tgclb  22028  0top  22041  en1top  22042  en2top  22043  topcld  22094  clsval2  22109  ntr0  22140  opnnei  22179  0nei  22187  restrcl  22216  rest0  22228  ordtrest2lem  22262  iocpnfordt  22274  icomnfordt  22275  cnindis  22351  isconn2  22473  kqtop  22804  mopn0  23560  locfinref  31693  ordtrest2NEWlem  31774  sxbrsigalem3  32139  cnambfre  35752