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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 5351 | . 2 ⊢ ∅ ∈ 𝒫 3o | |
2 | eqeq1 2729 | . . . . 5 ⊢ (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅})) | |
3 | id 22 | . . . . 5 ⊢ (𝑟 = ∅ → 𝑟 = ∅) | |
4 | 2, 3 | ifbieq2d 4551 | . . . 4 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(∅ = {∅}, {∅, 1o}, ∅)) |
5 | 0nep0 5353 | . . . . . . 7 ⊢ ∅ ≠ {∅} | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝑟 = ∅ → ∅ ≠ {∅}) |
7 | 6 | neneqd 2935 | . . . . 5 ⊢ (𝑟 = ∅ → ¬ ∅ = {∅}) |
8 | 7 | iffalsed 4536 | . . . 4 ⊢ (𝑟 = ∅ → if(∅ = {∅}, {∅, 1o}, ∅) = ∅) |
9 | 4, 8 | eqtrd 2765 | . . 3 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = ∅) |
10 | clsk1indlem.k | . . 3 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) | |
11 | 0ex 5303 | . . 3 ⊢ ∅ ∈ V | |
12 | 9, 10, 11 | fvmpt 6998 | . 2 ⊢ (∅ ∈ 𝒫 3o → (𝐾‘∅) = ∅) |
13 | 1, 12 | ax-mp 5 | 1 ⊢ (𝐾‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∅c0 4319 ifcif 4525 𝒫 cpw 4599 {csn 4625 {cpr 4627 ↦ cmpt 5227 ‘cfv 6543 1oc1o 8473 3oc3o 8475 |
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 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3944 df-un 3946 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-iota 6495 df-fun 6545 df-fv 6551 |
This theorem is referenced by: clsk1independent 43537 |
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