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Theorem sn0cld 21096
 Description: The closed sets of the topology {∅}. (Contributed by FL, 5-Jan-2009.)
Assertion
Ref Expression
sn0cld (Clsd‘{∅}) = {∅}

Proof of Theorem sn0cld
StepHypRef Expression
1 0ex 4942 . . 3 ∅ ∈ V
2 discld 21095 . . 3 (∅ ∈ V → (Clsd‘𝒫 ∅) = 𝒫 ∅)
31, 2ax-mp 5 . 2 (Clsd‘𝒫 ∅) = 𝒫 ∅
4 pw0 4488 . . 3 𝒫 ∅ = {∅}
54fveq2i 6355 . 2 (Clsd‘𝒫 ∅) = (Clsd‘{∅})
63, 5, 43eqtr3i 2790 1 (Clsd‘{∅}) = {∅}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ∈ wcel 2139  Vcvv 3340  ∅c0 4058  𝒫 cpw 4302  {csn 4321  ‘cfv 6049  Clsdccld 21022 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-top 20901  df-cld 21025 This theorem is referenced by: (None)
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