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Theorem clsk1indlem0 38945
Description: The ansatz closure function (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) has the K0 property of preserving the nullary union. (Contributed by RP, 6-Jul-2021.)
Hypothesis
Ref Expression
clsk1indlem.k 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
Assertion
Ref Expression
clsk1indlem0 (𝐾‘∅) = ∅

Proof of Theorem clsk1indlem0
StepHypRef Expression
1 0elpw 4992 . 2 ∅ ∈ 𝒫 3𝑜
2 eqeq1 2769 . . . . 5 (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅}))
3 id 22 . . . . 5 (𝑟 = ∅ → 𝑟 = ∅)
42, 3ifbieq2d 4268 . . . 4 (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(∅ = {∅}, {∅, 1𝑜}, ∅))
5 0nep0 4994 . . . . . . 7 ∅ ≠ {∅}
65a1i 11 . . . . . 6 (𝑟 = ∅ → ∅ ≠ {∅})
76neneqd 2942 . . . . 5 (𝑟 = ∅ → ¬ ∅ = {∅})
87iffalsed 4254 . . . 4 (𝑟 = ∅ → if(∅ = {∅}, {∅, 1𝑜}, ∅) = ∅)
94, 8eqtrd 2799 . . 3 (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = ∅)
10 clsk1indlem.k . . 3 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
11 0ex 4950 . . 3 ∅ ∈ V
129, 10, 11fvmpt 6471 . 2 (∅ ∈ 𝒫 3𝑜 → (𝐾‘∅) = ∅)
131, 12ax-mp 5 1 (𝐾‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  wcel 2155  wne 2937  c0 4079  ifcif 4243  𝒫 cpw 4315  {csn 4334  {cpr 4336  cmpt 4888  cfv 6068  1𝑜c1o 7757  3𝑜c3o 7759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076
This theorem is referenced by:  clsk1independent  38950
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