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Theorem cls0 21664
Description: The closure of the empty set. (Contributed by NM, 2-Oct-2007.) (Proof shortened by Jim Kingdon, 12-Mar-2023.)
Assertion
Ref Expression
cls0 (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)

Proof of Theorem cls0
StepHypRef Expression
1 0cld 21622 . 2 (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
2 cldcls 21626 . 2 (∅ ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘∅) = ∅)
31, 2syl 17 1 (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  c0 4269  cfv 6331  Topctop 21477  Clsdccld 21600  clsccl 21602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-top 21478  df-cld 21603  df-cls 21605
This theorem is referenced by:  dfac14lem  22201  flimclslem  22568
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