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Theorem clsk1indlem0 38866
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 4966 . 2 ∅ ∈ 𝒫 3𝑜
2 eqeq1 2775 . . . . 5 (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅}))
3 id 22 . . . . 5 (𝑟 = ∅ → 𝑟 = ∅)
42, 3ifbieq2d 4251 . . . 4 (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(∅ = {∅}, {∅, 1𝑜}, ∅))
5 0nep0 4968 . . . . . . 7 ∅ ≠ {∅}
65a1i 11 . . . . . 6 (𝑟 = ∅ → ∅ ≠ {∅})
76neneqd 2948 . . . . 5 (𝑟 = ∅ → ¬ ∅ = {∅})
87iffalsed 4237 . . . 4 (𝑟 = ∅ → if(∅ = {∅}, {∅, 1𝑜}, ∅) = ∅)
94, 8eqtrd 2805 . . 3 (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = ∅)
10 clsk1indlem.k . . 3 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
11 0ex 4925 . . 3 ∅ ∈ V
129, 10, 11fvmpt 6425 . 2 (∅ ∈ 𝒫 3𝑜 → (𝐾‘∅) = ∅)
131, 12ax-mp 5 1 (𝐾‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  wne 2943  c0 4064  ifcif 4226  𝒫 cpw 4298  {csn 4317  {cpr 4319  cmpt 4864  cfv 6032  1𝑜c1o 7707  3𝑜c3o 7709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5995  df-fun 6034  df-fv 6040
This theorem is referenced by:  clsk1independent  38871
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