Theorem clsk1indlem0 38339

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 4834	. 2 ⊢ ∅ ∈ 𝒫 3𝑜
2		eqeq1 2626	. . . . 5 ⊢ (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅}))
3		id 22	. . . . 5 ⊢ (𝑟 = ∅ → 𝑟 = ∅)
4	2, 3	ifbieq2d 4111	. . . 4 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(∅ = {∅}, {∅, 1𝑜}, ∅))
5		0nep0 4836	. . . . . . 7 ⊢ ∅ ≠ {∅}
6	5	a1i 11	. . . . . 6 ⊢ (𝑟 = ∅ → ∅ ≠ {∅})
7	6	neneqd 2799	. . . . 5 ⊢ (𝑟 = ∅ → ¬ ∅ = {∅})
8	7	iffalsed 4097	. . . 4 ⊢ (𝑟 = ∅ → if(∅ = {∅}, {∅, 1𝑜}, ∅) = ∅)
9	4, 8	eqtrd 2656	. . 3 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = ∅)
10		clsk1indlem.k	. . 3 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
11		0ex 4790	. . 3 ⊢ ∅ ∈ V
12	9, 10, 11	fvmpt 6282	. 2 ⊢ (∅ ∈ 𝒫 3𝑜 → (𝐾‘∅) = ∅)
13	1, 12	ax-mp 5	1 ⊢ (𝐾‘∅) = ∅

Syntax hints:   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  {csn 4177  {cpr 4179   ↦ cmpt 4729  ‘cfv 5888  1_𝑜c1o 7553  3_𝑜c3o 7555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  clsk1independent  38344