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Theorem clsk1indlem0 37359
 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 4760 . 2 ∅ ∈ 𝒫 3𝑜
2 eqeq1 2614 . . . . 5 (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅}))
3 id 22 . . . . 5 (𝑟 = ∅ → 𝑟 = ∅)
42, 3ifbieq2d 4061 . . . 4 (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(∅ = {∅}, {∅, 1𝑜}, ∅))
5 0nep0 4762 . . . . . . 7 ∅ ≠ {∅}
65a1i 11 . . . . . 6 (𝑟 = ∅ → ∅ ≠ {∅})
76neneqd 2787 . . . . 5 (𝑟 = ∅ → ¬ ∅ = {∅})
87iffalsed 4047 . . . 4 (𝑟 = ∅ → if(∅ = {∅}, {∅, 1𝑜}, ∅) = ∅)
94, 8eqtrd 2644 . . 3 (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = ∅)
10 clsk1indlem.k . . 3 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
11 0ex 4718 . . 3 ∅ ∈ V
129, 10, 11fvmpt 6191 . 2 (∅ ∈ 𝒫 3𝑜 → (𝐾‘∅) = ∅)
131, 12ax-mp 5 1 (𝐾‘∅) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  {csn 4125  {cpr 4127   ↦ cmpt 4643  ‘cfv 5804  1𝑜c1o 7440  3𝑜c3o 7442 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  clsk1independent  37364
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