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Theorem clsk1indlem2 38861
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 4491 . . . . . . . . . 10 {∅} ⊆ {∅, 1𝑜}
31, 2syl6eqss 3797 . . . . . . . . 9 (𝑠 = {∅} → 𝑠 ⊆ {∅, 1𝑜})
43ancli 575 . . . . . . . 8 (𝑠 = {∅} → (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}))
54con3i 150 . . . . . . 7 (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) → ¬ 𝑠 = {∅})
6 ssid 3766 . . . . . . 7 𝑠𝑠
75, 6jctir 562 . . . . . 6 (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) → (¬ 𝑠 = {∅} ∧ 𝑠𝑠))
87orri 390 . . . . 5 ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠))
98a1i 11 . . . 4 (𝑠 ∈ 𝒫 3𝑜 → ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠)))
10 sseq2 3769 . . . . 5 (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅, 1𝑜} → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ 𝑠 ⊆ {∅, 1𝑜}))
11 sseq2 3769 . . . . 5 (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = 𝑠 → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ 𝑠𝑠))
1210, 11elimif 4267 . . . 4 (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠)))
139, 12sylibr 224 . . 3 (𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
14 eqeq1 2765 . . . . 5 (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
15 id 22 . . . . 5 (𝑟 = 𝑠𝑟 = 𝑠)
1614, 15ifbieq2d 4256 . . . 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 5059 . . . . 5 {∅, 1𝑜} ∈ V
19 vex 3344 . . . . 5 𝑠 ∈ V
2018, 19ifex 4301 . . . 4 if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ V
2116, 17, 20fvmpt 6446 . . 3 (𝑠 ∈ 𝒫 3𝑜 → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
2213, 21sseqtr4d 3784 . 2 (𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝐾𝑠))
2322rgen 3061 1 𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝐾𝑠)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 382  wa 383   = wceq 1632  wcel 2140  wral 3051  wss 3716  c0 4059  ifcif 4231  𝒫 cpw 4303  {csn 4322  {cpr 4324  cmpt 4882  cfv 6050  1𝑜c1o 7724  3𝑜c3o 7726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-iota 6013  df-fun 6052  df-fv 6058
This theorem is referenced by:  clsk1independent  38865
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