Theorem clsk1indlem0 41651

Description: The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K0 property of preserving the nullary union. (Contributed by RP, 6-Jul-2021.)

Hypothesis

Ref	Expression
clsk1indlem.k	⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))

Assertion

Ref	Expression
clsk1indlem0	⊢ (𝐾‘∅) = ∅

Proof of Theorem clsk1indlem0

Step	Hyp	Ref	Expression
1		0elpw 5278	. 2 ⊢ ∅ ∈ 𝒫 3o
2		eqeq1 2742	. . . . 5 ⊢ (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅}))
3		id 22	. . . . 5 ⊢ (𝑟 = ∅ → 𝑟 = ∅)
4	2, 3	ifbieq2d 4485	. . . 4 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(∅ = {∅}, {∅, 1o}, ∅))
5		0nep0 5280	. . . . . . 7 ⊢ ∅ ≠ {∅}
6	5	a1i 11	. . . . . 6 ⊢ (𝑟 = ∅ → ∅ ≠ {∅})
7	6	neneqd 2948	. . . . 5 ⊢ (𝑟 = ∅ → ¬ ∅ = {∅})
8	7	iffalsed 4470	. . . 4 ⊢ (𝑟 = ∅ → if(∅ = {∅}, {∅, 1o}, ∅) = ∅)
9	4, 8	eqtrd 2778	. . 3 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = ∅)
10		clsk1indlem.k	. . 3 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
11		0ex 5231	. . 3 ⊢ ∅ ∈ V
12	9, 10, 11	fvmpt 6875	. 2 ⊢ (∅ ∈ 𝒫 3o → (𝐾‘∅) = ∅)
13	1, 12	ax-mp 5	1 ⊢ (𝐾‘∅) = ∅

Syntax hints:   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∅c0 4256  ifcif 4459  𝒫 cpw 4533  {csn 4561  {cpr 4563   ↦ cmpt 5157  ‘cfv 6433  1_oc1o 8290  3_oc3o 8292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441

This theorem is referenced by:  clsk1independent  41656