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Theorem clsk1indlem0 43384
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 5350 . 2 ∅ ∈ 𝒫 3o
2 eqeq1 2731 . . . . 5 (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅}))
3 id 22 . . . . 5 (𝑟 = ∅ → 𝑟 = ∅)
42, 3ifbieq2d 4550 . . . 4 (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(∅ = {∅}, {∅, 1o}, ∅))
5 0nep0 5352 . . . . . . 7 ∅ ≠ {∅}
65a1i 11 . . . . . 6 (𝑟 = ∅ → ∅ ≠ {∅})
76neneqd 2940 . . . . 5 (𝑟 = ∅ → ¬ ∅ = {∅})
87iffalsed 4535 . . . 4 (𝑟 = ∅ → if(∅ = {∅}, {∅, 1o}, ∅) = ∅)
94, 8eqtrd 2767 . . 3 (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = ∅)
10 clsk1indlem.k . . 3 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
11 0ex 5301 . . 3 ∅ ∈ V
129, 10, 11fvmpt 6999 . 2 (∅ ∈ 𝒫 3o → (𝐾‘∅) = ∅)
131, 12ax-mp 5 1 (𝐾‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  wne 2935  c0 4318  ifcif 4524  𝒫 cpw 4598  {csn 4624  {cpr 4626  cmpt 5225  cfv 6542  1oc1o 8471  3oc3o 8473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550
This theorem is referenced by:  clsk1independent  43389
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