Theorem clsk1indlem0 44787

Description: The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 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

Step	Hyp	Ref	Expression
1		0elpw 5326	. 2 ⊢ ∅ ∈ 𝒫 3o
2		eqeq1 2767	. . . . 5 ⊢ (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅}))
3		id 23	. . . . 5 ⊢ (𝑟 = ∅ → 𝑟 = ∅)
4	2, 3	ifbieq2d 4514	. . . 4 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(∅ = {∅}, {∅, 1o}, ∅))
5		0nep0 5328	. . . . . . 7 ⊢ ∅ ≠ {∅}
6	5	a1i 11	. . . . . 6 ⊢ (𝑟 = ∅ → ∅ ≠ {∅})
7	6	neneqd 2963	. . . . 5 ⊢ (𝑟 = ∅ → ¬ ∅ = {∅})
8	7	iffalsed 4498	. . . 4 ⊢ (𝑟 = ∅ → if(∅ = {∅}, {∅, 1o}, ∅) = ∅)
9	4, 8	eqtrd 2798	. . 3 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = ∅)
10		clsk1indlem.k	. . 3 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
11		0ex 5270	. . 3 ⊢ ∅ ∈ V
12	9, 10, 11	fvmpt 6989	. 2 ⊢ (∅ ∈ 𝒫 3o → (𝐾‘∅) = ∅)
13	1, 12	ax-mp 5	1 ⊢ (𝐾‘∅) = ∅

Syntax hints:   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∅c0 4286  ifcif 4487  𝒫 cpw 4562  {csn 4589  {cpr 4591   ↦ cmpt 5192  ‘cfv 6536  1_oc1o 8442  3_oc3o 8444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544

This theorem is referenced by:  clsk1independent  44792