Theorem clsk1indlem2 41652

Description: The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K2 property of expanding. (Contributed by RP, 6-Jul-2021.)

Hypothesis

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

Assertion

Ref	Expression
clsk1indlem2	⊢ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾‘𝑠)

Distinct variable group:   𝑠,𝑟

Allowed substitution hints:   𝐾(𝑠,𝑟)

Proof of Theorem clsk1indlem2

Step	Hyp	Ref	Expression
1		id 22	. . . . . . . . . 10 ⊢ (𝑠 = {∅} → 𝑠 = {∅})
2		snsspr1 4747	. . . . . . . . . 10 ⊢ {∅} ⊆ {∅, 1o}
3	1, 2	eqsstrdi 3975	. . . . . . . . 9 ⊢ (𝑠 = {∅} → 𝑠 ⊆ {∅, 1o})
4	3	ancli 549	. . . . . . . 8 ⊢ (𝑠 = {∅} → (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}))
5	4	con3i 154	. . . . . . 7 ⊢ (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → ¬ 𝑠 = {∅})
6		ssid 3943	. . . . . . 7 ⊢ 𝑠 ⊆ 𝑠
7	5, 6	jctir 521	. . . . . 6 ⊢ (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠))
8	7	orri 859	. . . . 5 ⊢ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠))
9	8	a1i 11	. . . 4 ⊢ (𝑠 ∈ 𝒫 3o → ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠)))
10		sseq2 3947	. . . . 5 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅, 1o} → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠 ⊆ {∅, 1o}))
11		sseq2 3947	. . . . 5 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = 𝑠 → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠 ⊆ 𝑠))
12	10, 11	elimif 4496	. . . 4 ⊢ (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠)))
13	9, 12	sylibr 233	. . 3 ⊢ (𝑠 ∈ 𝒫 3o → 𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠))
14		eqeq1 2742	. . . . 5 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
15		id 22	. . . . 5 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
16	14, 15	ifbieq2d 4485	. . . 4 ⊢ (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
17		clsk1indlem.k	. . . 4 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
18		prex 5355	. . . . 5 ⊢ {∅, 1o} ∈ V
19		vex 3436	. . . . 5 ⊢ 𝑠 ∈ V
20	18, 19	ifex 4509	. . . 4 ⊢ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
21	16, 17, 20	fvmpt 6875	. . 3 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
22	13, 21	sseqtrrd 3962	. 2 ⊢ (𝑠 ∈ 𝒫 3o → 𝑠 ⊆ (𝐾‘𝑠))
23	22	rgen 3074	1 ⊢ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾‘𝑠)

Syntax hints:  ¬ wn 3   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887  ∅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-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-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