Theorem clsk1indlem0 41540

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 5273	. 2 ⊢ ∅ ∈ 𝒫 3o
2		eqeq1 2742	. . . . 5 ⊢ (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅}))
3		id 22	. . . . 5 ⊢ (𝑟 = ∅ → 𝑟 = ∅)
4	2, 3	ifbieq2d 4482	. . . 4 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(∅ = {∅}, {∅, 1o}, ∅))
5		0nep0 5275	. . . . . . 7 ⊢ ∅ ≠ {∅}
6	5	a1i 11	. . . . . 6 ⊢ (𝑟 = ∅ → ∅ ≠ {∅})
7	6	neneqd 2947	. . . . 5 ⊢ (𝑟 = ∅ → ¬ ∅ = {∅})
8	7	iffalsed 4467	. . . 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 5226	. . 3 ⊢ ∅ ∈ V
12	9, 10, 11	fvmpt 6857	. 2 ⊢ (∅ ∈ 𝒫 3o → (𝐾‘∅) = ∅)
13	1, 12	ax-mp 5	1 ⊢ (𝐾‘∅) = ∅

Syntax hints:   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∅c0 4253  ifcif 4456  𝒫 cpw 4530  {csn 4558  {cpr 4560   ↦ cmpt 5153  ‘cfv 6418  1_oc1o 8260  3_oc3o 8262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426

This theorem is referenced by:  clsk1independent  41545