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Theorem clsk1indlem1 38862
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 7102 . . . . . 6 {∅, 1𝑜, 2𝑜} ∈ V
21a1i 11 . . . . 5 (⊤ → {∅, 1𝑜, 2𝑜} ∈ V)
3 snsstp1 4482 . . . . . 6 {∅} ⊆ {∅, 1𝑜, 2𝑜}
43a1i 11 . . . . 5 (⊤ → {∅} ⊆ {∅, 1𝑜, 2𝑜})
52, 4sselpwd 4941 . . . 4 (⊤ → {∅} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
65trud 1641 . . 3 {∅} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
7 df3o2 38841 . . . 4 3𝑜 = {∅, 1𝑜, 2𝑜}
87pweqi 4301 . . 3 𝒫 3𝑜 = 𝒫 {∅, 1𝑜, 2𝑜}
96, 8eleqtrri 2849 . 2 {∅} ∈ 𝒫 3𝑜
10 0ex 4924 . . . . . . . 8 ∅ ∈ V
1110snss 4451 . . . . . . 7 (∅ ∈ {∅, 1𝑜, 2𝑜} ↔ {∅} ⊆ {∅, 1𝑜, 2𝑜})
124, 11sylibr 224 . . . . . 6 (⊤ → ∅ ∈ {∅, 1𝑜, 2𝑜})
13 snsstp3 4484 . . . . . . . 8 {2𝑜} ⊆ {∅, 1𝑜, 2𝑜}
1413a1i 11 . . . . . . 7 (⊤ → {2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
15 2on 7720 . . . . . . . . 9 2𝑜 ∈ On
1615elexi 3365 . . . . . . . 8 2𝑜 ∈ V
1716snss 4451 . . . . . . 7 (2𝑜 ∈ {∅, 1𝑜, 2𝑜} ↔ {2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
1814, 17sylibr 224 . . . . . 6 (⊤ → 2𝑜 ∈ {∅, 1𝑜, 2𝑜})
1912, 18prssd 4488 . . . . 5 (⊤ → {∅, 2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
202, 19sselpwd 4941 . . . 4 (⊤ → {∅, 2𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
2120trud 1641 . . 3 {∅, 2𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
2221, 8eleqtrri 2849 . 2 {∅, 2𝑜} ∈ 𝒫 3𝑜
23 simpl 468 . . 3 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → {∅} ∈ 𝒫 3𝑜)
24 sseq1 3775 . . . . . 6 (𝑠 = {∅} → (𝑠𝑡 ↔ {∅} ⊆ 𝑡))
25 fveq2 6330 . . . . . . . 8 (𝑠 = {∅} → (𝐾𝑠) = (𝐾‘{∅}))
2625sseq1d 3781 . . . . . . 7 (𝑠 = {∅} → ((𝐾𝑠) ⊆ (𝐾𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
2726notbid 307 . . . . . 6 (𝑠 = {∅} → (¬ (𝐾𝑠) ⊆ (𝐾𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
2824, 27anbi12d 616 . . . . 5 (𝑠 = {∅} → ((𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
2928rexbidv 3200 . . . 4 (𝑠 = {∅} → (∃𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
3029adantl 467 . . 3 ((({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) ∧ 𝑠 = {∅}) → (∃𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
31 simpr 471 . . . 4 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → {∅, 2𝑜} ∈ 𝒫 3𝑜)
32 fveq2 6330 . . . . . . . 8 (𝑡 = {∅, 2𝑜} → (𝐾𝑡) = (𝐾‘{∅, 2𝑜}))
3332sseq2d 3782 . . . . . . 7 (𝑡 = {∅, 2𝑜} → ((𝐾‘{∅}) ⊆ (𝐾𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
3433notbid 307 . . . . . 6 (𝑡 = {∅, 2𝑜} → (¬ (𝐾‘{∅}) ⊆ (𝐾𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
3534cleq2lem 38433 . . . . 5 (𝑡 = {∅, 2𝑜} → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))))
3635adantl 467 . . . 4 ((({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) ∧ 𝑡 = {∅, 2𝑜}) → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))))
37 1oex 7719 . . . . . . . 8 1𝑜 ∈ V
3837prid2 4434 . . . . . . 7 1𝑜 ∈ {∅, 1𝑜}
39 iftrue 4231 . . . . . . . . 9 (𝑟 = {∅} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = {∅, 1𝑜})
40 clsk1indlem.k . . . . . . . . 9 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
41 prex 5037 . . . . . . . . 9 {∅, 1𝑜} ∈ V
4239, 40, 41fvmpt 6422 . . . . . . . 8 ({∅} ∈ 𝒫 3𝑜 → (𝐾‘{∅}) = {∅, 1𝑜})
4342adantr 466 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → (𝐾‘{∅}) = {∅, 1𝑜})
4438, 43syl5eleqr 2857 . . . . . 6 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → 1𝑜 ∈ (𝐾‘{∅}))
45 1n0 7727 . . . . . . . . . . 11 1𝑜 ≠ ∅
4645neii 2945 . . . . . . . . . 10 ¬ 1𝑜 = ∅
47 eqcom 2778 . . . . . . . . . . . 12 (1𝑜 = 2𝑜 ↔ 2𝑜 = 1𝑜)
48 df-2o 7712 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
49 df-1o 7711 . . . . . . . . . . . . 13 1𝑜 = suc ∅
5048, 49eqeq12i 2785 . . . . . . . . . . . 12 (2𝑜 = 1𝑜 ↔ suc 1𝑜 = suc ∅)
51 suc11reg 8678 . . . . . . . . . . . 12 (suc 1𝑜 = suc ∅ ↔ 1𝑜 = ∅)
5247, 50, 513bitri 286 . . . . . . . . . . 11 (1𝑜 = 2𝑜 ↔ 1𝑜 = ∅)
5345, 52nemtbir 3038 . . . . . . . . . 10 ¬ 1𝑜 = 2𝑜
5446, 53pm3.2ni 867 . . . . . . . . 9 ¬ (1𝑜 = ∅ ∨ 1𝑜 = 2𝑜)
55 elpri 4337 . . . . . . . . 9 (1𝑜 ∈ {∅, 2𝑜} → (1𝑜 = ∅ ∨ 1𝑜 = 2𝑜))
5654, 55mto 188 . . . . . . . 8 ¬ 1𝑜 ∈ {∅, 2𝑜}
5756a1i 11 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ 1𝑜 ∈ {∅, 2𝑜})
58 eqeq1 2775 . . . . . . . . . . 11 (𝑟 = {∅, 2𝑜} → (𝑟 = {∅} ↔ {∅, 2𝑜} = {∅}))
59 id 22 . . . . . . . . . . 11 (𝑟 = {∅, 2𝑜} → 𝑟 = {∅, 2𝑜})
6058, 59ifbieq2d 4250 . . . . . . . . . 10 (𝑟 = {∅, 2𝑜} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if({∅, 2𝑜} = {∅}, {∅, 1𝑜}, {∅, 2𝑜}))
6116prid2 4434 . . . . . . . . . . . 12 2𝑜 ∈ {∅, 2𝑜}
62 2on0 7721 . . . . . . . . . . . . 13 2𝑜 ≠ ∅
63 nelsn 4351 . . . . . . . . . . . . 13 (2𝑜 ≠ ∅ → ¬ 2𝑜 ∈ {∅})
6462, 63ax-mp 5 . . . . . . . . . . . 12 ¬ 2𝑜 ∈ {∅}
65 nelneq2 2875 . . . . . . . . . . . 12 ((2𝑜 ∈ {∅, 2𝑜} ∧ ¬ 2𝑜 ∈ {∅}) → ¬ {∅, 2𝑜} = {∅})
6661, 64, 65mp2an 672 . . . . . . . . . . 11 ¬ {∅, 2𝑜} = {∅}
6766iffalsei 4235 . . . . . . . . . 10 if({∅, 2𝑜} = {∅}, {∅, 1𝑜}, {∅, 2𝑜}) = {∅, 2𝑜}
6860, 67syl6eq 2821 . . . . . . . . 9 (𝑟 = {∅, 2𝑜} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = {∅, 2𝑜})
69 prex 5037 . . . . . . . . 9 {∅, 2𝑜} ∈ V
7068, 40, 69fvmpt 6422 . . . . . . . 8 ({∅, 2𝑜} ∈ 𝒫 3𝑜 → (𝐾‘{∅, 2𝑜}) = {∅, 2𝑜})
7170adantl 467 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → (𝐾‘{∅, 2𝑜}) = {∅, 2𝑜})
7257, 71neleqtrrd 2872 . . . . . 6 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ 1𝑜 ∈ (𝐾‘{∅, 2𝑜}))
73 nelss 3813 . . . . . 6 ((1𝑜 ∈ (𝐾‘{∅}) ∧ ¬ 1𝑜 ∈ (𝐾‘{∅, 2𝑜})) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))
7444, 72, 73syl2anc 573 . . . . 5 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))
75 snsspr1 4480 . . . . 5 {∅} ⊆ {∅, 2𝑜}
7674, 75jctil 509 . . . 4 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
7731, 36, 76rspcedvd 3467 . . 3 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
7823, 30, 77rspcedvd 3467 . 2 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ∃𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)))
799, 22, 78mp2an 672 1 𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  wo 836   = wceq 1631  wtru 1632  wcel 2145  wne 2943  wrex 3062  Vcvv 3351  wss 3723  c0 4063  ifcif 4225  𝒫 cpw 4297  {csn 4316  {cpr 4318  {ctp 4320  cmpt 4863  Oncon0 5864  suc csuc 5866  cfv 6029  1𝑜c1o 7704  2𝑜c2o 7705  3𝑜c3o 7706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7094  ax-reg 8651
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-ord 5867  df-on 5868  df-suc 5870  df-iota 5992  df-fun 6031  df-fv 6037  df-1o 7711  df-2o 7712  df-3o 7713
This theorem is referenced by:  clsk1independent  38863
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