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Theorem clsk1indlem2 40788
 Description: The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 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 ⊆ (𝐾𝑠)
Distinct variable group:   𝑠,𝑟
Allowed substitution hints:   𝐾(𝑠,𝑟)

Proof of Theorem clsk1indlem2
StepHypRef Expression
1 id 22 . . . . . . . . . 10 (𝑠 = {∅} → 𝑠 = {∅})
2 snsspr1 4707 . . . . . . . . . 10 {∅} ⊆ {∅, 1o}
31, 2eqsstrdi 3969 . . . . . . . . 9 (𝑠 = {∅} → 𝑠 ⊆ {∅, 1o})
43ancli 552 . . . . . . . 8 (𝑠 = {∅} → (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}))
54con3i 157 . . . . . . 7 (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → ¬ 𝑠 = {∅})
6 ssid 3937 . . . . . . 7 𝑠𝑠
75, 6jctir 524 . . . . . 6 (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → (¬ 𝑠 = {∅} ∧ 𝑠𝑠))
87orri 859 . . . . 5 ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠))
98a1i 11 . . . 4 (𝑠 ∈ 𝒫 3o → ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠)))
10 sseq2 3941 . . . . 5 (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅, 1o} → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠 ⊆ {∅, 1o}))
11 sseq2 3941 . . . . 5 (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = 𝑠 → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠𝑠))
1210, 11elimif 4461 . . . 4 (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠)))
139, 12sylibr 237 . . 3 (𝑠 ∈ 𝒫 3o𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠))
14 eqeq1 2802 . . . . 5 (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
15 id 22 . . . . 5 (𝑟 = 𝑠𝑟 = 𝑠)
1614, 15ifbieq2d 4450 . . . 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 5299 . . . . 5 {∅, 1o} ∈ V
19 vex 3444 . . . . 5 𝑠 ∈ V
2018, 19ifex 4473 . . . 4 if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
2116, 17, 20fvmpt 6746 . . 3 (𝑠 ∈ 𝒫 3o → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
2213, 21sseqtrrd 3956 . 2 (𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾𝑠))
2322rgen 3116 1 𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾𝑠)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881  ∅c0 4243  ifcif 4425  𝒫 cpw 4497  {csn 4525  {cpr 4527   ↦ cmpt 5111  ‘cfv 6325  1oc1o 8081  3oc3o 8083 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fv 6333 This theorem is referenced by:  clsk1independent  40792
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