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Theorem clsk1indlem1 39041
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 7159 . . . . . 6 {∅, 1𝑜, 2𝑜} ∈ V
21a1i 11 . . . . 5 (⊤ → {∅, 1𝑜, 2𝑜} ∈ V)
3 snsstp1 4503 . . . . . 6 {∅} ⊆ {∅, 1𝑜, 2𝑜}
43a1i 11 . . . . 5 (⊤ → {∅} ⊆ {∅, 1𝑜, 2𝑜})
52, 4sselpwd 4970 . . . 4 (⊤ → {∅} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
65mptru 1660 . . 3 {∅} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
7 df3o2 39020 . . . 4 3𝑜 = {∅, 1𝑜, 2𝑜}
87pweqi 4321 . . 3 𝒫 3𝑜 = 𝒫 {∅, 1𝑜, 2𝑜}
96, 8eleqtrri 2843 . 2 {∅} ∈ 𝒫 3𝑜
10 0ex 4952 . . . . . . . 8 ∅ ∈ V
1110snss 4472 . . . . . . 7 (∅ ∈ {∅, 1𝑜, 2𝑜} ↔ {∅} ⊆ {∅, 1𝑜, 2𝑜})
124, 11sylibr 225 . . . . . 6 (⊤ → ∅ ∈ {∅, 1𝑜, 2𝑜})
13 snsstp3 4505 . . . . . . . 8 {2𝑜} ⊆ {∅, 1𝑜, 2𝑜}
1413a1i 11 . . . . . . 7 (⊤ → {2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
15 2on 7777 . . . . . . . . 9 2𝑜 ∈ On
1615elexi 3366 . . . . . . . 8 2𝑜 ∈ V
1716snss 4472 . . . . . . 7 (2𝑜 ∈ {∅, 1𝑜, 2𝑜} ↔ {2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
1814, 17sylibr 225 . . . . . 6 (⊤ → 2𝑜 ∈ {∅, 1𝑜, 2𝑜})
1912, 18prssd 4509 . . . . 5 (⊤ → {∅, 2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
202, 19sselpwd 4970 . . . 4 (⊤ → {∅, 2𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
2120mptru 1660 . . 3 {∅, 2𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
2221, 8eleqtrri 2843 . 2 {∅, 2𝑜} ∈ 𝒫 3𝑜
23 simpl 474 . . 3 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → {∅} ∈ 𝒫 3𝑜)
24 sseq1 3788 . . . . . 6 (𝑠 = {∅} → (𝑠𝑡 ↔ {∅} ⊆ 𝑡))
25 fveq2 6379 . . . . . . . 8 (𝑠 = {∅} → (𝐾𝑠) = (𝐾‘{∅}))
2625sseq1d 3794 . . . . . . 7 (𝑠 = {∅} → ((𝐾𝑠) ⊆ (𝐾𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
2726notbid 309 . . . . . 6 (𝑠 = {∅} → (¬ (𝐾𝑠) ⊆ (𝐾𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
2824, 27anbi12d 624 . . . . 5 (𝑠 = {∅} → ((𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
2928rexbidv 3199 . . . 4 (𝑠 = {∅} → (∃𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
3029adantl 473 . . 3 ((({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) ∧ 𝑠 = {∅}) → (∃𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
31 simpr 477 . . . 4 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → {∅, 2𝑜} ∈ 𝒫 3𝑜)
32 fveq2 6379 . . . . . . . 8 (𝑡 = {∅, 2𝑜} → (𝐾𝑡) = (𝐾‘{∅, 2𝑜}))
3332sseq2d 3795 . . . . . . 7 (𝑡 = {∅, 2𝑜} → ((𝐾‘{∅}) ⊆ (𝐾𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
3433notbid 309 . . . . . 6 (𝑡 = {∅, 2𝑜} → (¬ (𝐾‘{∅}) ⊆ (𝐾𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
3534cleq2lem 38613 . . . . 5 (𝑡 = {∅, 2𝑜} → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))))
3635adantl 473 . . . 4 ((({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) ∧ 𝑡 = {∅, 2𝑜}) → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))))
37 1oex 7776 . . . . . . . 8 1𝑜 ∈ V
3837prid2 4455 . . . . . . 7 1𝑜 ∈ {∅, 1𝑜}
39 iftrue 4251 . . . . . . . . 9 (𝑟 = {∅} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = {∅, 1𝑜})
40 clsk1indlem.k . . . . . . . . 9 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
41 prex 5067 . . . . . . . . 9 {∅, 1𝑜} ∈ V
4239, 40, 41fvmpt 6475 . . . . . . . 8 ({∅} ∈ 𝒫 3𝑜 → (𝐾‘{∅}) = {∅, 1𝑜})
4342adantr 472 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → (𝐾‘{∅}) = {∅, 1𝑜})
4438, 43syl5eleqr 2851 . . . . . 6 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → 1𝑜 ∈ (𝐾‘{∅}))
45 1n0 7784 . . . . . . . . . . 11 1𝑜 ≠ ∅
4645neii 2939 . . . . . . . . . 10 ¬ 1𝑜 = ∅
47 eqcom 2772 . . . . . . . . . . . 12 (1𝑜 = 2𝑜 ↔ 2𝑜 = 1𝑜)
48 df-2o 7769 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
49 df-1o 7768 . . . . . . . . . . . . 13 1𝑜 = suc ∅
5048, 49eqeq12i 2779 . . . . . . . . . . . 12 (2𝑜 = 1𝑜 ↔ suc 1𝑜 = suc ∅)
51 suc11reg 8735 . . . . . . . . . . . 12 (suc 1𝑜 = suc ∅ ↔ 1𝑜 = ∅)
5247, 50, 513bitri 288 . . . . . . . . . . 11 (1𝑜 = 2𝑜 ↔ 1𝑜 = ∅)
5345, 52nemtbir 3032 . . . . . . . . . 10 ¬ 1𝑜 = 2𝑜
5446, 53pm3.2ni 904 . . . . . . . . 9 ¬ (1𝑜 = ∅ ∨ 1𝑜 = 2𝑜)
55 elpri 4358 . . . . . . . . 9 (1𝑜 ∈ {∅, 2𝑜} → (1𝑜 = ∅ ∨ 1𝑜 = 2𝑜))
5654, 55mto 188 . . . . . . . 8 ¬ 1𝑜 ∈ {∅, 2𝑜}
5756a1i 11 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ 1𝑜 ∈ {∅, 2𝑜})
58 eqeq1 2769 . . . . . . . . . . 11 (𝑟 = {∅, 2𝑜} → (𝑟 = {∅} ↔ {∅, 2𝑜} = {∅}))
59 id 22 . . . . . . . . . . 11 (𝑟 = {∅, 2𝑜} → 𝑟 = {∅, 2𝑜})
6058, 59ifbieq2d 4270 . . . . . . . . . 10 (𝑟 = {∅, 2𝑜} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if({∅, 2𝑜} = {∅}, {∅, 1𝑜}, {∅, 2𝑜}))
6116prid2 4455 . . . . . . . . . . . 12 2𝑜 ∈ {∅, 2𝑜}
62 2on0 7778 . . . . . . . . . . . . 13 2𝑜 ≠ ∅
63 nelsn 4372 . . . . . . . . . . . . 13 (2𝑜 ≠ ∅ → ¬ 2𝑜 ∈ {∅})
6462, 63ax-mp 5 . . . . . . . . . . . 12 ¬ 2𝑜 ∈ {∅}
65 nelneq2 2869 . . . . . . . . . . . 12 ((2𝑜 ∈ {∅, 2𝑜} ∧ ¬ 2𝑜 ∈ {∅}) → ¬ {∅, 2𝑜} = {∅})
6661, 64, 65mp2an 683 . . . . . . . . . . 11 ¬ {∅, 2𝑜} = {∅}
6766iffalsei 4255 . . . . . . . . . 10 if({∅, 2𝑜} = {∅}, {∅, 1𝑜}, {∅, 2𝑜}) = {∅, 2𝑜}
6860, 67syl6eq 2815 . . . . . . . . 9 (𝑟 = {∅, 2𝑜} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = {∅, 2𝑜})
69 prex 5067 . . . . . . . . 9 {∅, 2𝑜} ∈ V
7068, 40, 69fvmpt 6475 . . . . . . . 8 ({∅, 2𝑜} ∈ 𝒫 3𝑜 → (𝐾‘{∅, 2𝑜}) = {∅, 2𝑜})
7170adantl 473 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → (𝐾‘{∅, 2𝑜}) = {∅, 2𝑜})
7257, 71neleqtrrd 2866 . . . . . 6 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ 1𝑜 ∈ (𝐾‘{∅, 2𝑜}))
73 nelss 3826 . . . . . 6 ((1𝑜 ∈ (𝐾‘{∅}) ∧ ¬ 1𝑜 ∈ (𝐾‘{∅, 2𝑜})) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))
7444, 72, 73syl2anc 579 . . . . 5 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))
75 snsspr1 4501 . . . . 5 {∅} ⊆ {∅, 2𝑜}
7674, 75jctil 515 . . . 4 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
7731, 36, 76rspcedvd 3469 . . 3 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
7823, 30, 77rspcedvd 3469 . 2 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ∃𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)))
799, 22, 78mp2an 683 1 𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384  wo 873   = wceq 1652  wtru 1653  wcel 2155  wne 2937  wrex 3056  Vcvv 3350  wss 3734  c0 4081  ifcif 4245  𝒫 cpw 4317  {csn 4336  {cpr 4338  {ctp 4340  cmpt 4890  Oncon0 5910  suc csuc 5912  cfv 6070  1𝑜c1o 7761  2𝑜c2o 7762  3𝑜c3o 7763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064  ax-un 7151  ax-reg 8708
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-ord 5913  df-on 5914  df-suc 5916  df-iota 6033  df-fun 6072  df-fv 6078  df-1o 7768  df-2o 7769  df-3o 7770
This theorem is referenced by:  clsk1independent  39042
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