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Theorem clsk1indlem2 41329
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 4727 . . . . . . . . . 10 {∅} ⊆ {∅, 1o}
31, 2eqsstrdi 3955 . . . . . . . . 9 (𝑠 = {∅} → 𝑠 ⊆ {∅, 1o})
43ancli 552 . . . . . . . 8 (𝑠 = {∅} → (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}))
54con3i 157 . . . . . . 7 (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → ¬ 𝑠 = {∅})
6 ssid 3923 . . . . . . 7 𝑠𝑠
75, 6jctir 524 . . . . . 6 (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → (¬ 𝑠 = {∅} ∧ 𝑠𝑠))
87orri 862 . . . . 5 ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠))
98a1i 11 . . . 4 (𝑠 ∈ 𝒫 3o → ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠)))
10 sseq2 3927 . . . . 5 (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅, 1o} → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠 ⊆ {∅, 1o}))
11 sseq2 3927 . . . . 5 (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = 𝑠 → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠𝑠))
1210, 11elimif 4476 . . . 4 (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠)))
139, 12sylibr 237 . . 3 (𝑠 ∈ 𝒫 3o𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠))
14 eqeq1 2741 . . . . 5 (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
15 id 22 . . . . 5 (𝑟 = 𝑠𝑟 = 𝑠)
1614, 15ifbieq2d 4465 . . . 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 5325 . . . . 5 {∅, 1o} ∈ V
19 vex 3412 . . . . 5 𝑠 ∈ V
2018, 19ifex 4489 . . . 4 if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
2116, 17, 20fvmpt 6818 . . 3 (𝑠 ∈ 𝒫 3o → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
2213, 21sseqtrrd 3942 . 2 (𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾𝑠))
2322rgen 3071 1 𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾𝑠)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399  wo 847   = wceq 1543  wcel 2110  wral 3061  wss 3866  c0 4237  ifcif 4439  𝒫 cpw 4513  {csn 4541  {cpr 4543  cmpt 5135  cfv 6380  1oc1o 8195  3oc3o 8197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388
This theorem is referenced by:  clsk1independent  41333
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