Theorem clsk1indlem0 44045

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 5363	. 2 ⊢ ∅ ∈ 𝒫 3o
2		eqeq1 2740	. . . . 5 ⊢ (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅}))
3		id 22	. . . . 5 ⊢ (𝑟 = ∅ → 𝑟 = ∅)
4	2, 3	ifbieq2d 4558	. . . 4 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(∅ = {∅}, {∅, 1o}, ∅))
5		0nep0 5365	. . . . . . 7 ⊢ ∅ ≠ {∅}
6	5	a1i 11	. . . . . 6 ⊢ (𝑟 = ∅ → ∅ ≠ {∅})
7	6	neneqd 2944	. . . . 5 ⊢ (𝑟 = ∅ → ¬ ∅ = {∅})
8	7	iffalsed 4543	. . . 4 ⊢ (𝑟 = ∅ → if(∅ = {∅}, {∅, 1o}, ∅) = ∅)
9	4, 8	eqtrd 2776	. . 3 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = ∅)
10		clsk1indlem.k	. . 3 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
11		0ex 5314	. . 3 ⊢ ∅ ∈ V
12	9, 10, 11	fvmpt 7020	. 2 ⊢ (∅ ∈ 𝒫 3o → (𝐾‘∅) = ∅)
13	1, 12	ax-mp 5	1 ⊢ (𝐾‘∅) = ∅

Syntax hints:   = wceq 1538   ∈ wcel 2107   ≠ wne 2939  ∅c0 4340  ifcif 4532  𝒫 cpw 4606  {csn 4632  {cpr 4634   ↦ cmpt 5232  ‘cfv 6566  1_oc1o 8504  3_oc3o 8506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-nf 1782  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-ss 3981  df-nul 4341  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4914  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5584  df-xp 5696  df-rel 5697  df-cnv 5698  df-co 5699  df-dm 5700  df-iota 6519  df-fun 6568  df-fv 6574

This theorem is referenced by:  clsk1independent  44050