Theorem clsk1indlem0 40384

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 5248	. 2 ⊢ ∅ ∈ 𝒫 3o
2		eqeq1 2825	. . . . 5 ⊢ (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅}))
3		id 22	. . . . 5 ⊢ (𝑟 = ∅ → 𝑟 = ∅)
4	2, 3	ifbieq2d 4491	. . . 4 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(∅ = {∅}, {∅, 1o}, ∅))
5		0nep0 5250	. . . . . . 7 ⊢ ∅ ≠ {∅}
6	5	a1i 11	. . . . . 6 ⊢ (𝑟 = ∅ → ∅ ≠ {∅})
7	6	neneqd 3021	. . . . 5 ⊢ (𝑟 = ∅ → ¬ ∅ = {∅})
8	7	iffalsed 4477	. . . 4 ⊢ (𝑟 = ∅ → if(∅ = {∅}, {∅, 1o}, ∅) = ∅)
9	4, 8	eqtrd 2856	. . 3 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = ∅)
10		clsk1indlem.k	. . 3 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
11		0ex 5203	. . 3 ⊢ ∅ ∈ V
12	9, 10, 11	fvmpt 6762	. 2 ⊢ (∅ ∈ 𝒫 3o → (𝐾‘∅) = ∅)
13	1, 12	ax-mp 5	1 ⊢ (𝐾‘∅) = ∅

Syntax hints:   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∅c0 4290  ifcif 4466  𝒫 cpw 4538  {csn 4560  {cpr 4562   ↦ cmpt 5138  ‘cfv 6349  1_oc1o 8089  3_oc3o 8091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357

This theorem is referenced by:  clsk1independent  40389