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Mirrors > Home > MPE Home > Th. List > Mathboxes > clsk1indlem2 | Structured version Visualization version GIF version |
Description: The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) has the K2 property of expanding. (Contributed by RP, 6-Jul-2021.) |
Ref | Expression |
---|---|
clsk1indlem.k | ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) |
Ref | Expression |
---|---|
clsk1indlem2 | ⊢ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾‘𝑠) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . . . . . 10 ⊢ (𝑠 = {∅} → 𝑠 = {∅}) | |
2 | snsspr1 4818 | . . . . . . . . . 10 ⊢ {∅} ⊆ {∅, 1o} | |
3 | 1, 2 | eqsstrdi 4037 | . . . . . . . . 9 ⊢ (𝑠 = {∅} → 𝑠 ⊆ {∅, 1o}) |
4 | 3 | ancli 550 | . . . . . . . 8 ⊢ (𝑠 = {∅} → (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o})) |
5 | 4 | con3i 154 | . . . . . . 7 ⊢ (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → ¬ 𝑠 = {∅}) |
6 | ssid 4005 | . . . . . . 7 ⊢ 𝑠 ⊆ 𝑠 | |
7 | 5, 6 | jctir 522 | . . . . . 6 ⊢ (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠)) |
8 | 7 | orri 861 | . . . . 5 ⊢ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠)) |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝑠 ∈ 𝒫 3o → ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠))) |
10 | sseq2 4009 | . . . . 5 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅, 1o} → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠 ⊆ {∅, 1o})) | |
11 | sseq2 4009 | . . . . 5 ⊢ (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = 𝑠 → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠 ⊆ 𝑠)) | |
12 | 10, 11 | elimif 4566 | . . . 4 ⊢ (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠 ⊆ 𝑠))) |
13 | 9, 12 | sylibr 233 | . . 3 ⊢ (𝑠 ∈ 𝒫 3o → 𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠)) |
14 | eqeq1 2737 | . . . . 5 ⊢ (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅})) | |
15 | id 22 | . . . . 5 ⊢ (𝑟 = 𝑠 → 𝑟 = 𝑠) | |
16 | 14, 15 | ifbieq2d 4555 | . . . 4 ⊢ (𝑟 = 𝑠 → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(𝑠 = {∅}, {∅, 1o}, 𝑠)) |
17 | clsk1indlem.k | . . . 4 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) | |
18 | prex 5433 | . . . . 5 ⊢ {∅, 1o} ∈ V | |
19 | vex 3479 | . . . . 5 ⊢ 𝑠 ∈ V | |
20 | 18, 19 | ifex 4579 | . . . 4 ⊢ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V |
21 | 16, 17, 20 | fvmpt 6999 | . . 3 ⊢ (𝑠 ∈ 𝒫 3o → (𝐾‘𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠)) |
22 | 13, 21 | sseqtrrd 4024 | . 2 ⊢ (𝑠 ∈ 𝒫 3o → 𝑠 ⊆ (𝐾‘𝑠)) |
23 | 22 | rgen 3064 | 1 ⊢ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾‘𝑠) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ⊆ wss 3949 ∅c0 4323 ifcif 4529 𝒫 cpw 4603 {csn 4629 {cpr 4631 ↦ cmpt 5232 ‘cfv 6544 1oc1o 8459 3oc3o 8461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 |
This theorem is referenced by: clsk1independent 42797 |
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