Theorem clsk1indlem2 37360

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 4285	. . . . . . . . . 10 ⊢ {∅} ⊆ {∅, 1𝑜}
3	1, 2	syl6eqss 3618	. . . . . . . . 9 ⊢ (𝑠 = {∅} → 𝑠 ⊆ {∅, 1𝑜})
4	3	ancli 572	. . . . . . . 8 ⊢ (𝑠 = {∅} → (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}))
5	4	con3i 149	. . . . . . 7 ⊢ (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) → ¬ 𝑠 = {∅})
6		ssid 3587	. . . . . . 7 ⊢ 𝑠 ⊆ 𝑠
7	5, 6	jctir 559	. . . . . 6 ⊢ (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) → (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠))
8	7	orri 390	. . . . 5 ⊢ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠))
9	8	a1i 11	. . . 4 ⊢ (𝑠 ∈ 𝒫 3𝑜 → ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠)))
10		sseq2 3590	. . . . 5 ⊢ (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅, 1𝑜} → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ 𝑠 ⊆ {∅, 1𝑜}))
11		sseq2 3590	. . . . 5 ⊢ (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = 𝑠 → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ 𝑠 ⊆ 𝑠))
12	10, 11	elimif 4072	. . . 4 ⊢ (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠)))
13	9, 12	sylibr 223	. . 3 ⊢ (𝑠 ∈ 𝒫 3𝑜 → 𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
14		eqeq1 2614	. . . . 5 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
15		id 22	. . . . 5 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
16	14, 15	ifbieq2d 4061	. . . 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 4836	. . . . 5 ⊢ {∅, 1𝑜} ∈ V
19		vex 3176	. . . . 5 ⊢ 𝑠 ∈ V
20	18, 19	ifex 4106	. . . 4 ⊢ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ V
21	16, 17, 20	fvmpt 6191	. . 3 ⊢ (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
22	13, 21	sseqtr4d 3605	. 2 ⊢ (𝑠 ∈ 𝒫 3𝑜 → 𝑠 ⊆ (𝐾‘𝑠))
23	22	rgen 2906	1 ⊢ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝐾‘𝑠)

Syntax hints:  ¬ wn 3   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  {csn 4125  {cpr 4127   ↦ cmpt 4643  ‘cfv 5804  1_𝑜c1o 7440  3_𝑜c3o 7442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812

This theorem is referenced by:  clsk1independent  37364