Theorem clsk1indlem2 45179

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 23	. . . . . . . . . 10 ⊢ (𝑠 = {∅} → 𝑠 = {∅})
2		snsspr1 4769	. . . . . . . . . 10 ⊢ {∅} ⊆ {∅, 1o}
3	1, 2	eqsstrdi 3953	. . . . . . . . 9 ⊢ (𝑠 = {∅} → 𝑠 ⊆ {∅, 1o})
4	3	ancli 558	. . . . . . . 8 ⊢ (𝑠 = {∅} → (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}))
5	4	con3i 155	. . . . . . 7 ⊢ (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → ¬ 𝑠 = {∅})
6		ssid 3931	. . . . . . 7 ⊢ 𝑠 ⊆ 𝑠
7	5, 6	jctir 530	. . . . . 6 ⊢ (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠))
8	7	orri 876	. . . . 5 ⊢ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠))
9	8	a1i 11	. . . 4 ⊢ (𝑠 ∈ 𝒫 3o → ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠)))
10		sseq2 3935	. . . . 5 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅, 1o} → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠 ⊆ {∅, 1o}))
11		sseq2 3935	. . . . 5 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = 𝑠 → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠 ⊆ 𝑠))
12	10, 11	elimif 4519	. . . 4 ⊢ (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠)))
13	9, 12	sylibr 237	. . 3 ⊢ (𝑠 ∈ 𝒫 3o → 𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠))
14		eqeq1 2764	. . . . 5 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
15		id 23	. . . . 5 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠)
16	14, 15	ifbieq2d 4508	. . . 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 5396	. . . . 5 ⊢ {∅, 1o} ∈ V
19		vex 3454	. . . . 5 ⊢ 𝑠 ∈ V
20	18, 19	ifex 4532	. . . 4 ⊢ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
21	16, 17, 20	fvmpt 6998	. . 3 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
22	13, 21	sseqtrrd 3946	. 2 ⊢ (𝑠 ∈ 𝒫 3o → 𝑠 ⊆ (𝐾‘𝑠))
23	22	rgen 3078	1 ⊢ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾‘𝑠)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3896  ∅c0 4277  ifcif 4481  𝒫 cpw 4556  {csn 4584  {cpr 4586   ↦ cmpt 5185  ‘cfv 6543  1_oc1o 8477  3_oc3o 8479

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6497  df-fun 6545  df-fv 6551

This theorem is used by:  clsk1independent  45183