Theorem clsk1indlem2 44033

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 4795	. . . . . . . . . 10 ⊢ {∅} ⊆ {∅, 1o}
3	1, 2	eqsstrdi 4008	. . . . . . . . 9 ⊢ (𝑠 = {∅} → 𝑠 ⊆ {∅, 1o})
4	3	ancli 548	. . . . . . . 8 ⊢ (𝑠 = {∅} → (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}))
5	4	con3i 154	. . . . . . 7 ⊢ (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → ¬ 𝑠 = {∅})
6		ssid 3986	. . . . . . 7 ⊢ 𝑠 ⊆ 𝑠
7	5, 6	jctir 520	. . . . . 6 ⊢ (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠))
8	7	orri 862	. . . . 5 ⊢ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠))
9	8	a1i 11	. . . 4 ⊢ (𝑠 ∈ 𝒫 3o → ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠)))
10		sseq2 3990	. . . . 5 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅, 1o} → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠 ⊆ {∅, 1o}))
11		sseq2 3990	. . . . 5 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = 𝑠 → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠 ⊆ 𝑠))
12	10, 11	elimif 4543	. . . 4 ⊢ (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠)))
13	9, 12	sylibr 234	. . 3 ⊢ (𝑠 ∈ 𝒫 3o → 𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠))
14		eqeq1 2740	. . . . 5 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
15		id 22	. . . . 5 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
16	14, 15	ifbieq2d 4532	. . . 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 5412	. . . . 5 ⊢ {∅, 1o} ∈ V
19		vex 3468	. . . . 5 ⊢ 𝑠 ∈ V
20	18, 19	ifex 4556	. . . 4 ⊢ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
21	16, 17, 20	fvmpt 6991	. . 3 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
22	13, 21	sseqtrrd 4001	. 2 ⊢ (𝑠 ∈ 𝒫 3o → 𝑠 ⊆ (𝐾‘𝑠))
23	22	rgen 3054	1 ⊢ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾‘𝑠)

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3052   ⊆ wss 3931  ∅c0 4313  ifcif 4505  𝒫 cpw 4580  {csn 4606  {cpr 4608   ↦ cmpt 5206  ‘cfv 6536  1_oc1o 8478  3_oc3o 8480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544

This theorem is referenced by:  clsk1independent  44037