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Theorem clsk1indlem0 40253
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 5252 . 2 ∅ ∈ 𝒫 3o
2 eqeq1 2828 . . . . 5 (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅}))
3 id 22 . . . . 5 (𝑟 = ∅ → 𝑟 = ∅)
42, 3ifbieq2d 4494 . . . 4 (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(∅ = {∅}, {∅, 1o}, ∅))
5 0nep0 5254 . . . . . . 7 ∅ ≠ {∅}
65a1i 11 . . . . . 6 (𝑟 = ∅ → ∅ ≠ {∅})
76neneqd 3025 . . . . 5 (𝑟 = ∅ → ¬ ∅ = {∅})
87iffalsed 4480 . . . 4 (𝑟 = ∅ → if(∅ = {∅}, {∅, 1o}, ∅) = ∅)
94, 8eqtrd 2860 . . 3 (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = ∅)
10 clsk1indlem.k . . 3 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
11 0ex 5207 . . 3 ∅ ∈ V
129, 10, 11fvmpt 6764 . 2 (∅ ∈ 𝒫 3o → (𝐾‘∅) = ∅)
131, 12ax-mp 5 1 (𝐾‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2106  wne 3020  c0 4294  ifcif 4469  𝒫 cpw 4541  {csn 4563  {cpr 4565  cmpt 5142  cfv 6351  1oc1o 8089  3oc3o 8091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fv 6359
This theorem is referenced by:  clsk1independent  40258
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