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Theorem clsk1indlem2 42436
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 4779 . . . . . . . . . 10 {∅} ⊆ {∅, 1o}
31, 2eqsstrdi 4001 . . . . . . . . 9 (𝑠 = {∅} → 𝑠 ⊆ {∅, 1o})
43ancli 549 . . . . . . . 8 (𝑠 = {∅} → (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}))
54con3i 154 . . . . . . 7 (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → ¬ 𝑠 = {∅})
6 ssid 3969 . . . . . . 7 𝑠𝑠
75, 6jctir 521 . . . . . 6 (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → (¬ 𝑠 = {∅} ∧ 𝑠𝑠))
87orri 860 . . . . 5 ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠))
98a1i 11 . . . 4 (𝑠 ∈ 𝒫 3o → ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠)))
10 sseq2 3973 . . . . 5 (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅, 1o} → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠 ⊆ {∅, 1o}))
11 sseq2 3973 . . . . 5 (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = 𝑠 → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠𝑠))
1210, 11elimif 4528 . . . 4 (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠)))
139, 12sylibr 233 . . 3 (𝑠 ∈ 𝒫 3o𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠))
14 eqeq1 2735 . . . . 5 (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
15 id 22 . . . . 5 (𝑟 = 𝑠𝑟 = 𝑠)
1614, 15ifbieq2d 4517 . . . 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 5394 . . . . 5 {∅, 1o} ∈ V
19 vex 3450 . . . . 5 𝑠 ∈ V
2018, 19ifex 4541 . . . 4 if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
2116, 17, 20fvmpt 6953 . . 3 (𝑠 ∈ 𝒫 3o → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
2213, 21sseqtrrd 3988 . 2 (𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾𝑠))
2322rgen 3062 1 𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾𝑠)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wo 845   = wceq 1541  wcel 2106  wral 3060  wss 3913  c0 4287  ifcif 4491  𝒫 cpw 4565  {csn 4591  {cpr 4593  cmpt 5193  cfv 6501  1oc1o 8410  3oc3o 8412
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509
This theorem is referenced by:  clsk1independent  42440
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