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Theorem sn0cld 21626
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 5202 . . 3 ∅ ∈ V
2 discld 21625 . . 3 (∅ ∈ V → (Clsd‘𝒫 ∅) = 𝒫 ∅)
31, 2ax-mp 5 . 2 (Clsd‘𝒫 ∅) = 𝒫 ∅
4 pw0 4737 . . 3 𝒫 ∅ = {∅}
54fveq2i 6666 . 2 (Clsd‘𝒫 ∅) = (Clsd‘{∅})
63, 5, 43eqtr3i 2849 1 (Clsd‘{∅}) = {∅}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  Vcvv 3492  c0 4288  𝒫 cpw 4535  {csn 4557  cfv 6348  Clsdccld 21552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-top 21430  df-cld 21555
This theorem is referenced by: (None)
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