Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clsk1indlem2 Structured version   Visualization version   GIF version

Theorem clsk1indlem2 42294
Description: The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) has the K2 property of expanding. (Contributed by RP, 6-Jul-2021.)
Hypothesis
Ref Expression
clsk1indlem.k 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
Assertion
Ref Expression
clsk1indlem2 𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾𝑠)
Distinct variable group:   𝑠,𝑟
Allowed substitution hints:   𝐾(𝑠,𝑟)

Proof of Theorem clsk1indlem2
StepHypRef Expression
1 id 22 . . . . . . . . . 10 (𝑠 = {∅} → 𝑠 = {∅})
2 snsspr1 4773 . . . . . . . . . 10 {∅} ⊆ {∅, 1o}
31, 2eqsstrdi 3997 . . . . . . . . 9 (𝑠 = {∅} → 𝑠 ⊆ {∅, 1o})
43ancli 549 . . . . . . . 8 (𝑠 = {∅} → (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}))
54con3i 154 . . . . . . 7 (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → ¬ 𝑠 = {∅})
6 ssid 3965 . . . . . . 7 𝑠𝑠
75, 6jctir 521 . . . . . 6 (¬ (𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) → (¬ 𝑠 = {∅} ∧ 𝑠𝑠))
87orri 860 . . . . 5 ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠))
98a1i 11 . . . 4 (𝑠 ∈ 𝒫 3o → ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠)))
10 sseq2 3969 . . . . 5 (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = {∅, 1o} → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠 ⊆ {∅, 1o}))
11 sseq2 3969 . . . . 5 (if(𝑠 = {∅}, {∅, 1o}, 𝑠) = 𝑠 → (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ 𝑠𝑠))
1210, 11elimif 4522 . . . 4 (𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠) ↔ ((𝑠 = {∅} ∧ 𝑠 ⊆ {∅, 1o}) ∨ (¬ 𝑠 = {∅} ∧ 𝑠𝑠)))
139, 12sylibr 233 . . 3 (𝑠 ∈ 𝒫 3o𝑠 ⊆ if(𝑠 = {∅}, {∅, 1o}, 𝑠))
14 eqeq1 2740 . . . . 5 (𝑟 = 𝑠 → (𝑟 = {∅} ↔ 𝑠 = {∅}))
15 id 22 . . . . 5 (𝑟 = 𝑠𝑟 = 𝑠)
1614, 15ifbieq2d 4511 . . . 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 5388 . . . . 5 {∅, 1o} ∈ V
19 vex 3448 . . . . 5 𝑠 ∈ V
2018, 19ifex 4535 . . . 4 if(𝑠 = {∅}, {∅, 1o}, 𝑠) ∈ V
2116, 17, 20fvmpt 6946 . . 3 (𝑠 ∈ 𝒫 3o → (𝐾𝑠) = if(𝑠 = {∅}, {∅, 1o}, 𝑠))
2213, 21sseqtrrd 3984 . 2 (𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾𝑠))
2322rgen 3065 1 𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾𝑠)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wo 845   = wceq 1541  wcel 2106  wral 3063  wss 3909  c0 4281  ifcif 4485  𝒫 cpw 4559  {csn 4585  {cpr 4587  cmpt 5187  cfv 6494  1oc1o 8402  3oc3o 8404
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6446  df-fun 6496  df-fv 6502
This theorem is referenced by:  clsk1independent  42298
  Copyright terms: Public domain W3C validator