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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 5221 | . 2 ⊢ ∅ ∈ 𝒫 3o | |
2 | eqeq1 2802 | . . . . 5 ⊢ (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅})) | |
3 | id 22 | . . . . 5 ⊢ (𝑟 = ∅ → 𝑟 = ∅) | |
4 | 2, 3 | ifbieq2d 4450 | . . . 4 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(∅ = {∅}, {∅, 1o}, ∅)) |
5 | 0nep0 5223 | . . . . . . 7 ⊢ ∅ ≠ {∅} | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝑟 = ∅ → ∅ ≠ {∅}) |
7 | 6 | neneqd 2992 | . . . . 5 ⊢ (𝑟 = ∅ → ¬ ∅ = {∅}) |
8 | 7 | iffalsed 4436 | . . . 4 ⊢ (𝑟 = ∅ → if(∅ = {∅}, {∅, 1o}, ∅) = ∅) |
9 | 4, 8 | eqtrd 2833 | . . 3 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = ∅) |
10 | clsk1indlem.k | . . 3 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) | |
11 | 0ex 5175 | . . 3 ⊢ ∅ ∈ V | |
12 | 9, 10, 11 | fvmpt 6745 | . 2 ⊢ (∅ ∈ 𝒫 3o → (𝐾‘∅) = ∅) |
13 | 1, 12 | ax-mp 5 | 1 ⊢ (𝐾‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 ifcif 4425 𝒫 cpw 4497 {csn 4525 {cpr 4527 ↦ cmpt 5110 ‘cfv 6324 1oc1o 8078 3oc3o 8080 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: clsk1independent 40749 |
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