Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clsk1indlem1 Structured version   Visualization version   GIF version

Theorem clsk1indlem1 40275
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 7458 . . . 4 {∅, 1o, 2o} ∈ V
2 snsstp1 4743 . . . 4 {∅} ⊆ {∅, 1o, 2o}
31, 2elpwi2 5241 . . 3 {∅} ∈ 𝒫 {∅, 1o, 2o}
4 df3o2 40254 . . . 4 3o = {∅, 1o, 2o}
54pweqi 4541 . . 3 𝒫 3o = 𝒫 {∅, 1o, 2o}
63, 5eleqtrri 2912 . 2 {∅} ∈ 𝒫 3o
71a1i 11 . . . . 5 (⊤ → {∅, 1o, 2o} ∈ V)
82a1i 11 . . . . . . 7 (⊤ → {∅} ⊆ {∅, 1o, 2o})
9 0ex 5203 . . . . . . . 8 ∅ ∈ V
109snss 4712 . . . . . . 7 (∅ ∈ {∅, 1o, 2o} ↔ {∅} ⊆ {∅, 1o, 2o})
118, 10sylibr 235 . . . . . 6 (⊤ → ∅ ∈ {∅, 1o, 2o})
12 snsstp3 4745 . . . . . . . 8 {2o} ⊆ {∅, 1o, 2o}
1312a1i 11 . . . . . . 7 (⊤ → {2o} ⊆ {∅, 1o, 2o})
14 2oex 8103 . . . . . . . 8 2o ∈ V
1514snss 4712 . . . . . . 7 (2o ∈ {∅, 1o, 2o} ↔ {2o} ⊆ {∅, 1o, 2o})
1613, 15sylibr 235 . . . . . 6 (⊤ → 2o ∈ {∅, 1o, 2o})
1711, 16prssd 4749 . . . . 5 (⊤ → {∅, 2o} ⊆ {∅, 1o, 2o})
187, 17sselpwd 5222 . . . 4 (⊤ → {∅, 2o} ∈ 𝒫 {∅, 1o, 2o})
1918mptru 1535 . . 3 {∅, 2o} ∈ 𝒫 {∅, 1o, 2o}
2019, 5eleqtrri 2912 . 2 {∅, 2o} ∈ 𝒫 3o
21 simpl 483 . . 3 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → {∅} ∈ 𝒫 3o)
22 sseq1 3991 . . . . . 6 (𝑠 = {∅} → (𝑠𝑡 ↔ {∅} ⊆ 𝑡))
23 fveq2 6664 . . . . . . . 8 (𝑠 = {∅} → (𝐾𝑠) = (𝐾‘{∅}))
2423sseq1d 3997 . . . . . . 7 (𝑠 = {∅} → ((𝐾𝑠) ⊆ (𝐾𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
2524notbid 319 . . . . . 6 (𝑠 = {∅} → (¬ (𝐾𝑠) ⊆ (𝐾𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
2622, 25anbi12d 630 . . . . 5 (𝑠 = {∅} → ((𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
2726rexbidv 3297 . . . 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 6664 . . . . . . . 8 (𝑡 = {∅, 2o} → (𝐾𝑡) = (𝐾‘{∅, 2o}))
3130sseq2d 3998 . . . . . . 7 (𝑡 = {∅, 2o} → ((𝐾‘{∅}) ⊆ (𝐾𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o})))
3231notbid 319 . . . . . 6 (𝑡 = {∅, 2o} → (¬ (𝐾‘{∅}) ⊆ (𝐾𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o})))
3332cleq2lem 39848 . . . . 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 8101 . . . . . . . 8 1o ∈ V
3635prid2 4693 . . . . . . 7 1o ∈ {∅, 1o}
37 iftrue 4471 . . . . . . . . 9 (𝑟 = {∅} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = {∅, 1o})
38 clsk1indlem.k . . . . . . . . 9 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
39 prex 5324 . . . . . . . . 9 {∅, 1o} ∈ V
4037, 38, 39fvmpt 6762 . . . . . . . 8 ({∅} ∈ 𝒫 3o → (𝐾‘{∅}) = {∅, 1o})
4140adantr 481 . . . . . . 7 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → (𝐾‘{∅}) = {∅, 1o})
4236, 41eleqtrrid 2920 . . . . . 6 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → 1o ∈ (𝐾‘{∅}))
43 1n0 8110 . . . . . . . . . . 11 1o ≠ ∅
4443neii 3018 . . . . . . . . . 10 ¬ 1o = ∅
45 eqcom 2828 . . . . . . . . . . . 12 (1o = 2o ↔ 2o = 1o)
46 df-2o 8094 . . . . . . . . . . . . 13 2o = suc 1o
47 df-1o 8093 . . . . . . . . . . . . 13 1o = suc ∅
4846, 47eqeq12i 2836 . . . . . . . . . . . 12 (2o = 1o ↔ suc 1o = suc ∅)
49 suc11reg 9071 . . . . . . . . . . . 12 (suc 1o = suc ∅ ↔ 1o = ∅)
5045, 48, 493bitri 298 . . . . . . . . . . 11 (1o = 2o ↔ 1o = ∅)
5143, 50nemtbir 3112 . . . . . . . . . 10 ¬ 1o = 2o
5244, 51pm3.2ni 874 . . . . . . . . 9 ¬ (1o = ∅ ∨ 1o = 2o)
53 elpri 4581 . . . . . . . . 9 (1o ∈ {∅, 2o} → (1o = ∅ ∨ 1o = 2o))
5452, 53mto 198 . . . . . . . 8 ¬ 1o ∈ {∅, 2o}
5554a1i 11 . . . . . . 7 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ 1o ∈ {∅, 2o})
56 eqeq1 2825 . . . . . . . . . . 11 (𝑟 = {∅, 2o} → (𝑟 = {∅} ↔ {∅, 2o} = {∅}))
57 id 22 . . . . . . . . . . 11 (𝑟 = {∅, 2o} → 𝑟 = {∅, 2o})
5856, 57ifbieq2d 4490 . . . . . . . . . 10 (𝑟 = {∅, 2o} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if({∅, 2o} = {∅}, {∅, 1o}, {∅, 2o}))
5914prid2 4693 . . . . . . . . . . . 12 2o ∈ {∅, 2o}
60 2on0 8104 . . . . . . . . . . . . 13 2o ≠ ∅
61 nelsn 4597 . . . . . . . . . . . . 13 (2o ≠ ∅ → ¬ 2o ∈ {∅})
6260, 61ax-mp 5 . . . . . . . . . . . 12 ¬ 2o ∈ {∅}
63 nelneq2 2938 . . . . . . . . . . . 12 ((2o ∈ {∅, 2o} ∧ ¬ 2o ∈ {∅}) → ¬ {∅, 2o} = {∅})
6459, 62, 63mp2an 688 . . . . . . . . . . 11 ¬ {∅, 2o} = {∅}
6564iffalsei 4475 . . . . . . . . . 10 if({∅, 2o} = {∅}, {∅, 1o}, {∅, 2o}) = {∅, 2o}
6658, 65syl6eq 2872 . . . . . . . . 9 (𝑟 = {∅, 2o} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = {∅, 2o})
67 prex 5324 . . . . . . . . 9 {∅, 2o} ∈ V
6866, 38, 67fvmpt 6762 . . . . . . . 8 ({∅, 2o} ∈ 𝒫 3o → (𝐾‘{∅, 2o}) = {∅, 2o})
6968adantl 482 . . . . . . 7 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → (𝐾‘{∅, 2o}) = {∅, 2o})
7055, 69neleqtrrd 2935 . . . . . 6 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ 1o ∈ (𝐾‘{∅, 2o}))
71 nelss 4029 . . . . . 6 ((1o ∈ (𝐾‘{∅}) ∧ ¬ 1o ∈ (𝐾‘{∅, 2o})) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))
7242, 70, 71syl2anc 584 . . . . 5 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))
73 snsspr1 4741 . . . . 5 {∅} ⊆ {∅, 2o}
7472, 73jctil 520 . . . 4 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ({∅} ⊆ {∅, 2o} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o})))
7529, 34, 74rspcedvd 3625 . . 3 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ∃𝑡 ∈ 𝒫 3o({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
7621, 28, 75rspcedvd 3625 . 2 (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ∃𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)))
776, 20, 76mp2an 688 1 𝑠 ∈ 𝒫 3o𝑡 ∈ 𝒫 3o(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396  wo 841   = wceq 1528  wtru 1529  wcel 2105  wne 3016  wrex 3139  Vcvv 3495  wss 3935  c0 4290  ifcif 4465  𝒫 cpw 4537  {csn 4559  {cpr 4561  {ctp 4563  cmpt 5138  suc csuc 6187  cfv 6349  1oc1o 8086  2oc2o 8087  3oc3o 8088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7450  ax-reg 9045
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-ord 6188  df-on 6189  df-suc 6191  df-iota 6308  df-fun 6351  df-fv 6357  df-1o 8093  df-2o 8094  df-3o 8095
This theorem is referenced by:  clsk1independent  40276
  Copyright terms: Public domain W3C validator