Theorem sn0cld 23098

Description: The closed sets of the topology {∅}. (Contributed by FL, 5-Jan-2009.)

Assertion

Ref	Expression
sn0cld	⊢ (Clsd‘{∅}) = {∅}

Proof of Theorem sn0cld

Step	Hyp	Ref	Expression
1		0ex 5307	. . 3 ⊢ ∅ ∈ V
2		discld 23097	. . 3 ⊢ (∅ ∈ V → (Clsd‘𝒫 ∅) = 𝒫 ∅)
3	1, 2	ax-mp 5	. 2 ⊢ (Clsd‘𝒫 ∅) = 𝒫 ∅
4		pw0 4812	. . 3 ⊢ 𝒫 ∅ = {∅}
5	4	fveq2i 6909	. 2 ⊢ (Clsd‘𝒫 ∅) = (Clsd‘{∅})
6	3, 5, 4	3eqtr3i 2773	1 ⊢ (Clsd‘{∅}) = {∅}

Syntax hints:   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ∅c0 4333  𝒫 cpw 4600  {csn 4626  ‘cfv 6561  Clsdccld 23024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-top 22900  df-cld 23027

This theorem is referenced by: (None)