Theorem clsk1indlem2 38340

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 4345	. . . . . . . . . 10 ⊢ {∅} ⊆ {∅, 1𝑜}
3	1, 2	syl6eqss 3655	. . . . . . . . 9 ⊢ (𝑠 = {∅} → 𝑠 ⊆ {∅, 1𝑜})
4	3	ancli 574	. . . . . . . 8 ⊢ (𝑠 = {∅} → (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}))
5	4	con3i 150	. . . . . . 7 ⊢ (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) → ¬ 𝑠 = {∅})
6		ssid 3624	. . . . . . 7 ⊢ 𝑠 ⊆ 𝑠
7	5, 6	jctir 561	. . . . . 6 ⊢ (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) → (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠))
8	7	orri 391	. . . . 5 ⊢ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠))
9	8	a1i 11	. . . 4 ⊢ (𝑠 ∈ 𝒫 3𝑜 → ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠)))
10		sseq2 3627	. . . . 5 ⊢ (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅, 1𝑜} → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ 𝑠 ⊆ {∅, 1𝑜}))
11		sseq2 3627	. . . . 5 ⊢ (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = 𝑠 → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ 𝑠 ⊆ 𝑠))
12	10, 11	elimif 4122	. . . 4 ⊢ (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠)))
13	9, 12	sylibr 224	. . 3 ⊢ (𝑠 ∈ 𝒫 3𝑜 → 𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
14		eqeq1 2626	. . . . 5 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
15		id 22	. . . . 5 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
16	14, 15	ifbieq2d 4111	. . . 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 4909	. . . . 5 ⊢ {∅, 1𝑜} ∈ V
19		vex 3203	. . . . 5 ⊢ 𝑠 ∈ V
20	18, 19	ifex 4156	. . . 4 ⊢ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ V
21	16, 17, 20	fvmpt 6282	. . 3 ⊢ (𝑠 ∈ 𝒫 3𝑜 → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
22	13, 21	sseqtr4d 3642	. 2 ⊢ (𝑠 ∈ 𝒫 3𝑜 → 𝑠 ⊆ (𝐾‘𝑠))
23	22	rgen 2922	1 ⊢ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝐾‘𝑠)

Syntax hints:  ¬ wn 3   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  ∅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-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-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