Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > clsk1indlem0 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
clsk1indlem.k | ⊢ 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) |
Ref | Expression |
---|---|
clsk1indlem0 | ⊢ (𝐾‘∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elpw 4760 | . 2 ⊢ ∅ ∈ 𝒫 3𝑜 | |
2 | eqeq1 2614 | . . . . 5 ⊢ (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅})) | |
3 | id 22 | . . . . 5 ⊢ (𝑟 = ∅ → 𝑟 = ∅) | |
4 | 2, 3 | ifbieq2d 4061 | . . . 4 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if(∅ = {∅}, {∅, 1𝑜}, ∅)) |
5 | 0nep0 4762 | . . . . . . 7 ⊢ ∅ ≠ {∅} | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝑟 = ∅ → ∅ ≠ {∅}) |
7 | 6 | neneqd 2787 | . . . . 5 ⊢ (𝑟 = ∅ → ¬ ∅ = {∅}) |
8 | 7 | iffalsed 4047 | . . . 4 ⊢ (𝑟 = ∅ → if(∅ = {∅}, {∅, 1𝑜}, ∅) = ∅) |
9 | 4, 8 | eqtrd 2644 | . . 3 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = ∅) |
10 | clsk1indlem.k | . . 3 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) | |
11 | 0ex 4718 | . . 3 ⊢ ∅ ∈ V | |
12 | 9, 10, 11 | fvmpt 6191 | . 2 ⊢ (∅ ∈ 𝒫 3𝑜 → (𝐾‘∅) = ∅) |
13 | 1, 12 | ax-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 |
Copyright terms: Public domain | W3C validator |