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Theorem clsk1indlem2 37857
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 4318 . . . . . . . . . 10 {∅} ⊆ {∅, 1𝑜}
31, 2syl6eqss 3639 . . . . . . . . 9 (𝑠 = {∅} → 𝑠 ⊆ {∅, 1𝑜})
43ancli 573 . . . . . . . 8 (𝑠 = {∅} → (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}))
54con3i 150 . . . . . . 7 (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) → ¬ 𝑠 = {∅})
6 ssid 3608 . . . . . . 7 𝑠𝑠
75, 6jctir 560 . . . . . 6 (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) → (¬ 𝑠 = {∅} ∧ 𝑠𝑠))
87orri 391 . . . . 5 ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠))
98a1i 11 . . . 4 (𝑠 ∈ 𝒫 3𝑜 → ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠)))
10 sseq2 3611 . . . . 5 (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = {∅, 1𝑜} → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ 𝑠 ⊆ {∅, 1𝑜}))
11 sseq2 3611 . . . . 5 (if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) = 𝑠 → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ 𝑠𝑠))
1210, 11elimif 4099 . . . 4 (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1𝑜}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠)))
139, 12sylibr 224 . . 3 (𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
14 eqeq1 2625 . . . . 5 (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
15 id 22 . . . . 5 (𝑟 = 𝑠𝑟 = 𝑠)
1614, 15ifbieq2d 4088 . . . 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 4875 . . . . 5 {∅, 1𝑜} ∈ V
19 vex 3192 . . . . 5 𝑠 ∈ V
2018, 19ifex 4133 . . . 4 if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠) ∈ V
2116, 17, 20fvmpt 6244 . . 3 (𝑠 ∈ 𝒫 3𝑜 → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1𝑜}, 𝑠))
2213, 21sseqtr4d 3626 . 2 (𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝐾𝑠))
2322rgen 2917 1 𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝐾𝑠)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 383  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3559  c0 3896  ifcif 4063  𝒫 cpw 4135  {csn 4153  {cpr 4155  cmpt 4678  cfv 5852  1𝑜c1o 7505  3𝑜c3o 7507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860
This theorem is referenced by:  clsk1independent  37861
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