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Theorem clsk1indlem1 42781
Description: The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) does not have the K1 property of isotony. (Contributed by RP, 6-Jul-2021.)
Hypothesis
Ref Expression
clsk1indlem.k 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
Assertion
Ref Expression
clsk1indlem1 𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡))
Distinct variable groups:   𝐾,𝑠,𝑡   𝑠,𝑟,𝑡
Allowed substitution hint:   𝐾(𝑟)

Proof of Theorem clsk1indlem1
StepHypRef Expression
1 tpex 7730 . . . 4 {∅, 1o, 2o} ∈ V
2 snsstp1 4818 . . . 4 {∅} ⊆ {∅, 1o, 2o}
31, 2elpwi2 5345 . . 3 {∅} ∈ 𝒫 {∅, 1o, 2o}
4 df3o2 42048 . . . 4 3o = {∅, 1o, 2o}
54pweqi 4617 . . 3 𝒫 3o = 𝒫 {∅, 1o, 2o}
63, 5eleqtrri 2832 . 2 {∅} ∈ 𝒫 3o
71a1i 11 . . . . 5 (⊤ → {∅, 1o, 2o} ∈ V)
82a1i 11 . . . . . . 7 (⊤ → {∅} ⊆ {∅, 1o, 2o})
9 0ex 5306 . . . . . . . 8 ∅ ∈ V
109snss 4788 . . . . . . 7 (∅ ∈ {∅, 1o, 2o} ↔ {∅} ⊆ {∅, 1o, 2o})
118, 10sylibr 233 . . . . . 6 (⊤ → ∅ ∈ {∅, 1o, 2o})
12 snsstp3 4820 . . . . . . . 8 {2o} ⊆ {∅, 1o, 2o}
1312a1i 11 . . . . . . 7 (⊤ → {2o} ⊆ {∅, 1o, 2o})
14 2oex 8473 . . . . . . . 8 2o ∈ V
1514snss 4788 . . . . . . 7 (2o ∈ {∅, 1o, 2o} ↔ {2o} ⊆ {∅, 1o, 2o})
1613, 15sylibr 233 . . . . . 6 (⊤ → 2o ∈ {∅, 1o, 2o})
1711, 16prssd 4824 . . . . 5 (⊤ → {∅, 2o} ⊆ {∅, 1o, 2o})
187, 17sselpwd 5325 . . . 4 (⊤ → {∅, 2o} ∈ 𝒫 {∅, 1o, 2o})
1918mptru 1548 . . 3 {∅, 2o} ∈ 𝒫 {∅, 1o, 2o}
2019, 5eleqtrri 2832 . 2 {∅, 2o} ∈ 𝒫 3o
21 simpl 483 . . 3 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → {∅} ∈ 𝒫 3o)
22 sseq1 4006 . . . . . 6 (𝑠 = {∅} → (𝑠𝑡 ↔ {∅} ⊆ 𝑡))
23 fveq2 6888 . . . . . . . 8 (𝑠 = {∅} → (𝐾𝑠) = (𝐾‘{∅}))
2423sseq1d 4012 . . . . . . 7 (𝑠 = {∅} → ((𝐾𝑠) ⊆ (𝐾𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
2524notbid 317 . . . . . 6 (𝑠 = {∅} → (¬ (𝐾𝑠) ⊆ (𝐾𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
2622, 25anbi12d 631 . . . . 5 (𝑠 = {∅} → ((𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
2726rexbidv 3178 . . . 4 (𝑠 = {∅} → (∃𝑡 ∈ 𝒫 3o(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ∃𝑡 ∈ 𝒫 3o({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
2827adantl 482 . . 3 ((({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) ∧ 𝑠 = {∅}) → (∃𝑡 ∈ 𝒫 3o(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ∃𝑡 ∈ 𝒫 3o({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
29 simpr 485 . . . 4 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → {∅, 2o} ∈ 𝒫 3o)
30 fveq2 6888 . . . . . . . 8 (𝑡 = {∅, 2o} → (𝐾𝑡) = (𝐾‘{∅, 2o}))
3130sseq2d 4013 . . . . . . 7 (𝑡 = {∅, 2o} → ((𝐾‘{∅}) ⊆ (𝐾𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o})))
3231notbid 317 . . . . . 6 (𝑡 = {∅, 2o} → (¬ (𝐾‘{∅}) ⊆ (𝐾𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o})))
3332cleq2lem 42344 . . . . 5 (𝑡 = {∅, 2o} → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ {∅, 2o} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))))
3433adantl 482 . . . 4 ((({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) ∧ 𝑡 = {∅, 2o}) → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ {∅, 2o} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))))
35 1oex 8472 . . . . . . . 8 1o ∈ V
3635prid2 4766 . . . . . . 7 1o ∈ {∅, 1o}
37 iftrue 4533 . . . . . . . . 9 (𝑟 = {∅} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = {∅, 1o})
38 clsk1indlem.k . . . . . . . . 9 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
39 prex 5431 . . . . . . . . 9 {∅, 1o} ∈ V
4037, 38, 39fvmpt 6995 . . . . . . . 8 ({∅} ∈ 𝒫 3o → (𝐾‘{∅}) = {∅, 1o})
4140adantr 481 . . . . . . 7 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → (𝐾‘{∅}) = {∅, 1o})
4236, 41eleqtrrid 2840 . . . . . 6 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → 1o ∈ (𝐾‘{∅}))
43 1n0 8484 . . . . . . . . . . 11 1o ≠ ∅
4443neii 2942 . . . . . . . . . 10 ¬ 1o = ∅
45 eqcom 2739 . . . . . . . . . . . 12 (1o = 2o ↔ 2o = 1o)
46 df-2o 8463 . . . . . . . . . . . . 13 2o = suc 1o
47 df-1o 8462 . . . . . . . . . . . . 13 1o = suc ∅
4846, 47eqeq12i 2750 . . . . . . . . . . . 12 (2o = 1o ↔ suc 1o = suc ∅)
49 suc11reg 9610 . . . . . . . . . . . 12 (suc 1o = suc ∅ ↔ 1o = ∅)
5045, 48, 493bitri 296 . . . . . . . . . . 11 (1o = 2o ↔ 1o = ∅)
5143, 50nemtbir 3038 . . . . . . . . . 10 ¬ 1o = 2o
5244, 51pm3.2ni 879 . . . . . . . . 9 ¬ (1o = ∅ ∨ 1o = 2o)
53 elpri 4649 . . . . . . . . 9 (1o ∈ {∅, 2o} → (1o = ∅ ∨ 1o = 2o))
5452, 53mto 196 . . . . . . . 8 ¬ 1o ∈ {∅, 2o}
5554a1i 11 . . . . . . 7 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ 1o ∈ {∅, 2o})
56 eqeq1 2736 . . . . . . . . . . 11 (𝑟 = {∅, 2o} → (𝑟 = {∅} ↔ {∅, 2o} = {∅}))
57 id 22 . . . . . . . . . . 11 (𝑟 = {∅, 2o} → 𝑟 = {∅, 2o})
5856, 57ifbieq2d 4553 . . . . . . . . . 10 (𝑟 = {∅, 2o} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if({∅, 2o} = {∅}, {∅, 1o}, {∅, 2o}))
5914prid2 4766 . . . . . . . . . . . 12 2o ∈ {∅, 2o}
60 2on0 8478 . . . . . . . . . . . . 13 2o ≠ ∅
61 nelsn 4667 . . . . . . . . . . . . 13 (2o ≠ ∅ → ¬ 2o ∈ {∅})
6260, 61ax-mp 5 . . . . . . . . . . . 12 ¬ 2o ∈ {∅}
63 nelneq2 2858 . . . . . . . . . . . 12 ((2o ∈ {∅, 2o} ∧ ¬ 2o ∈ {∅}) → ¬ {∅, 2o} = {∅})
6459, 62, 63mp2an 690 . . . . . . . . . . 11 ¬ {∅, 2o} = {∅}
6564iffalsei 4537 . . . . . . . . . 10 if({∅, 2o} = {∅}, {∅, 1o}, {∅, 2o}) = {∅, 2o}
6658, 65eqtrdi 2788 . . . . . . . . 9 (𝑟 = {∅, 2o} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = {∅, 2o})
67 prex 5431 . . . . . . . . 9 {∅, 2o} ∈ V
6866, 38, 67fvmpt 6995 . . . . . . . 8 ({∅, 2o} ∈ 𝒫 3o → (𝐾‘{∅, 2o}) = {∅, 2o})
6968adantl 482 . . . . . . 7 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → (𝐾‘{∅, 2o}) = {∅, 2o})
7055, 69neleqtrrd 2856 . . . . . 6 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ 1o ∈ (𝐾‘{∅, 2o}))
71 nelss 4046 . . . . . 6 ((1o ∈ (𝐾‘{∅}) ∧ ¬ 1o ∈ (𝐾‘{∅, 2o})) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))
7242, 70, 71syl2anc 584 . . . . 5 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))
73 snsspr1 4816 . . . . 5 {∅} ⊆ {∅, 2o}
7472, 73jctil 520 . . . 4 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ({∅} ⊆ {∅, 2o} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o})))
7529, 34, 74rspcedvd 3614 . . 3 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ∃𝑡 ∈ 𝒫 3o({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
7621, 28, 75rspcedvd 3614 . 2 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ∃𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)))
776, 20, 76mp2an 690 1 𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396  wo 845   = wceq 1541  wtru 1542  wcel 2106  wne 2940  wrex 3070  Vcvv 3474  wss 3947  c0 4321  ifcif 4527  𝒫 cpw 4601  {csn 4627  {cpr 4629  {ctp 4631  cmpt 5230  suc csuc 6363  cfv 6540  1oc1o 8455  2oc2o 8456  3oc3o 8457
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 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721  ax-reg 9583
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-eprel 5579  df-fr 5630  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-suc 6367  df-iota 6492  df-fun 6542  df-fv 6548  df-1o 8462  df-2o 8463  df-3o 8464
This theorem is referenced by:  clsk1independent  42782
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