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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 5350 | . 2 ⊢ ∅ ∈ 𝒫 3o | |
2 | eqeq1 2731 | . . . . 5 ⊢ (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅})) | |
3 | id 22 | . . . . 5 ⊢ (𝑟 = ∅ → 𝑟 = ∅) | |
4 | 2, 3 | ifbieq2d 4550 | . . . 4 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(∅ = {∅}, {∅, 1o}, ∅)) |
5 | 0nep0 5352 | . . . . . . 7 ⊢ ∅ ≠ {∅} | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ (𝑟 = ∅ → ∅ ≠ {∅}) |
7 | 6 | neneqd 2940 | . . . . 5 ⊢ (𝑟 = ∅ → ¬ ∅ = {∅}) |
8 | 7 | iffalsed 4535 | . . . 4 ⊢ (𝑟 = ∅ → if(∅ = {∅}, {∅, 1o}, ∅) = ∅) |
9 | 4, 8 | eqtrd 2767 | . . 3 ⊢ (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = ∅) |
10 | clsk1indlem.k | . . 3 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) | |
11 | 0ex 5301 | . . 3 ⊢ ∅ ∈ V | |
12 | 9, 10, 11 | fvmpt 6999 | . 2 ⊢ (∅ ∈ 𝒫 3o → (𝐾‘∅) = ∅) |
13 | 1, 12 | ax-mp 5 | 1 ⊢ (𝐾‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∅c0 4318 ifcif 4524 𝒫 cpw 4598 {csn 4624 {cpr 4626 ↦ cmpt 5225 ‘cfv 6542 1oc1o 8471 3oc3o 8473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 |
This theorem is referenced by: clsk1independent 43389 |
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