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Theorem 0cld 20652
Description: The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 4-Oct-2006.)
Assertion
Ref Expression
0cld (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))

Proof of Theorem 0cld
StepHypRef Expression
1 dif0 3904 . . 3 ( 𝐽 ∖ ∅) = 𝐽
21topopn 20536 . 2 (𝐽 ∈ Top → ( 𝐽 ∖ ∅) ∈ 𝐽)
3 0ss 3924 . . 3 ∅ ⊆ 𝐽
4 eqid 2610 . . . 4 𝐽 = 𝐽
54iscld2 20642 . . 3 ((𝐽 ∈ Top ∧ ∅ ⊆ 𝐽) → (∅ ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ ∅) ∈ 𝐽))
63, 5mpan2 703 . 2 (𝐽 ∈ Top → (∅ ∈ (Clsd‘𝐽) ↔ ( 𝐽 ∖ ∅) ∈ 𝐽))
72, 6mpbird 246 1 (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wcel 1977  cdif 3537  wss 3540  c0 3874   cuni 4372  cfv 5804  Topctop 20517  Clsdccld 20630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-top 20521  df-cld 20633
This theorem is referenced by:  cls0  20694  indiscld  20705  iscldtop  20709  iccordt  20828  iscon2  21027  tgptsmscld  21764  mblfinlem2  32617  mblfinlem3  32618  ismblfin  32620
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