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Theorem clsk1indlem1 37860
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 6917 . . . . . 6 {∅, 1𝑜, 2𝑜} ∈ V
21a1i 11 . . . . 5 (⊤ → {∅, 1𝑜, 2𝑜} ∈ V)
3 snsstp1 4320 . . . . . 6 {∅} ⊆ {∅, 1𝑜, 2𝑜}
43a1i 11 . . . . 5 (⊤ → {∅} ⊆ {∅, 1𝑜, 2𝑜})
52, 4sselpwd 4772 . . . 4 (⊤ → {∅} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
65trud 1490 . . 3 {∅} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
7 df3o2 37839 . . . 4 3𝑜 = {∅, 1𝑜, 2𝑜}
87pweqi 4139 . . 3 𝒫 3𝑜 = 𝒫 {∅, 1𝑜, 2𝑜}
96, 8eleqtrri 2697 . 2 {∅} ∈ 𝒫 3𝑜
10 0ex 4755 . . . . . . . 8 ∅ ∈ V
1110snss 4291 . . . . . . 7 (∅ ∈ {∅, 1𝑜, 2𝑜} ↔ {∅} ⊆ {∅, 1𝑜, 2𝑜})
124, 11sylibr 224 . . . . . 6 (⊤ → ∅ ∈ {∅, 1𝑜, 2𝑜})
13 snsstp3 4322 . . . . . . . 8 {2𝑜} ⊆ {∅, 1𝑜, 2𝑜}
1413a1i 11 . . . . . . 7 (⊤ → {2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
15 2on 7520 . . . . . . . . 9 2𝑜 ∈ On
1615elexi 3202 . . . . . . . 8 2𝑜 ∈ V
1716snss 4291 . . . . . . 7 (2𝑜 ∈ {∅, 1𝑜, 2𝑜} ↔ {2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
1814, 17sylibr 224 . . . . . 6 (⊤ → 2𝑜 ∈ {∅, 1𝑜, 2𝑜})
1912, 18prssd 4327 . . . . 5 (⊤ → {∅, 2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
202, 19sselpwd 4772 . . . 4 (⊤ → {∅, 2𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
2120trud 1490 . . 3 {∅, 2𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
2221, 8eleqtrri 2697 . 2 {∅, 2𝑜} ∈ 𝒫 3𝑜
23 simpl 473 . . 3 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → {∅} ∈ 𝒫 3𝑜)
24 sseq1 3610 . . . . . 6 (𝑠 = {∅} → (𝑠𝑡 ↔ {∅} ⊆ 𝑡))
25 fveq2 6153 . . . . . . . 8 (𝑠 = {∅} → (𝐾𝑠) = (𝐾‘{∅}))
2625sseq1d 3616 . . . . . . 7 (𝑠 = {∅} → ((𝐾𝑠) ⊆ (𝐾𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
2726notbid 308 . . . . . 6 (𝑠 = {∅} → (¬ (𝐾𝑠) ⊆ (𝐾𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
2824, 27anbi12d 746 . . . . 5 (𝑠 = {∅} → ((𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
2928rexbidv 3046 . . . 4 (𝑠 = {∅} → (∃𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
3029adantl 482 . . 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 6153 . . . . . . . 8 (𝑡 = {∅, 2𝑜} → (𝐾𝑡) = (𝐾‘{∅, 2𝑜}))
3332sseq2d 3617 . . . . . . 7 (𝑡 = {∅, 2𝑜} → ((𝐾‘{∅}) ⊆ (𝐾𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
3433notbid 308 . . . . . 6 (𝑡 = {∅, 2𝑜} → (¬ (𝐾‘{∅}) ⊆ (𝐾𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
3534cleq2lem 37430 . . . . 5 (𝑡 = {∅, 2𝑜} → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))))
3635adantl 482 . . . 4 ((({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) ∧ 𝑡 = {∅, 2𝑜}) → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))))
37 1on 7519 . . . . . . . . 9 1𝑜 ∈ On
3837elexi 3202 . . . . . . . 8 1𝑜 ∈ V
3938prid2 4273 . . . . . . 7 1𝑜 ∈ {∅, 1𝑜}
40 iftrue 4069 . . . . . . . . 9 (𝑟 = {∅} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = {∅, 1𝑜})
41 clsk1indlem.k . . . . . . . . 9 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
42 prex 4875 . . . . . . . . 9 {∅, 1𝑜} ∈ V
4340, 41, 42fvmpt 6244 . . . . . . . 8 ({∅} ∈ 𝒫 3𝑜 → (𝐾‘{∅}) = {∅, 1𝑜})
4443adantr 481 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → (𝐾‘{∅}) = {∅, 1𝑜})
4539, 44syl5eleqr 2705 . . . . . 6 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → 1𝑜 ∈ (𝐾‘{∅}))
46 1n0 7527 . . . . . . . . . . 11 1𝑜 ≠ ∅
4746neii 2792 . . . . . . . . . 10 ¬ 1𝑜 = ∅
48 eqcom 2628 . . . . . . . . . . . 12 (1𝑜 = 2𝑜 ↔ 2𝑜 = 1𝑜)
49 df-2o 7513 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
50 df-1o 7512 . . . . . . . . . . . . 13 1𝑜 = suc ∅
5149, 50eqeq12i 2635 . . . . . . . . . . . 12 (2𝑜 = 1𝑜 ↔ suc 1𝑜 = suc ∅)
52 suc11reg 8468 . . . . . . . . . . . 12 (suc 1𝑜 = suc ∅ ↔ 1𝑜 = ∅)
5348, 51, 523bitri 286 . . . . . . . . . . 11 (1𝑜 = 2𝑜 ↔ 1𝑜 = ∅)
5446, 53nemtbir 2885 . . . . . . . . . 10 ¬ 1𝑜 = 2𝑜
5547, 54pm3.2ni 898 . . . . . . . . 9 ¬ (1𝑜 = ∅ ∨ 1𝑜 = 2𝑜)
56 elpri 4173 . . . . . . . . 9 (1𝑜 ∈ {∅, 2𝑜} → (1𝑜 = ∅ ∨ 1𝑜 = 2𝑜))
5755, 56mto 188 . . . . . . . 8 ¬ 1𝑜 ∈ {∅, 2𝑜}
5857a1i 11 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ 1𝑜 ∈ {∅, 2𝑜})
59 eqeq1 2625 . . . . . . . . . . 11 (𝑟 = {∅, 2𝑜} → (𝑟 = {∅} ↔ {∅, 2𝑜} = {∅}))
60 id 22 . . . . . . . . . . 11 (𝑟 = {∅, 2𝑜} → 𝑟 = {∅, 2𝑜})
6159, 60ifbieq2d 4088 . . . . . . . . . 10 (𝑟 = {∅, 2𝑜} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if({∅, 2𝑜} = {∅}, {∅, 1𝑜}, {∅, 2𝑜}))
6216prid2 4273 . . . . . . . . . . . 12 2𝑜 ∈ {∅, 2𝑜}
63 2on0 7521 . . . . . . . . . . . . 13 2𝑜 ≠ ∅
64 nelsn 4188 . . . . . . . . . . . . 13 (2𝑜 ≠ ∅ → ¬ 2𝑜 ∈ {∅})
6563, 64ax-mp 5 . . . . . . . . . . . 12 ¬ 2𝑜 ∈ {∅}
66 nelneq2 2723 . . . . . . . . . . . 12 ((2𝑜 ∈ {∅, 2𝑜} ∧ ¬ 2𝑜 ∈ {∅}) → ¬ {∅, 2𝑜} = {∅})
6762, 65, 66mp2an 707 . . . . . . . . . . 11 ¬ {∅, 2𝑜} = {∅}
6867iffalsei 4073 . . . . . . . . . 10 if({∅, 2𝑜} = {∅}, {∅, 1𝑜}, {∅, 2𝑜}) = {∅, 2𝑜}
6961, 68syl6eq 2671 . . . . . . . . 9 (𝑟 = {∅, 2𝑜} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = {∅, 2𝑜})
70 prex 4875 . . . . . . . . 9 {∅, 2𝑜} ∈ V
7169, 41, 70fvmpt 6244 . . . . . . . 8 ({∅, 2𝑜} ∈ 𝒫 3𝑜 → (𝐾‘{∅, 2𝑜}) = {∅, 2𝑜})
7271adantl 482 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → (𝐾‘{∅, 2𝑜}) = {∅, 2𝑜})
7358, 72neleqtrrd 2720 . . . . . 6 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ 1𝑜 ∈ (𝐾‘{∅, 2𝑜}))
74 nelss 3648 . . . . . 6 ((1𝑜 ∈ (𝐾‘{∅}) ∧ ¬ 1𝑜 ∈ (𝐾‘{∅, 2𝑜})) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))
7545, 73, 74syl2anc 692 . . . . 5 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))
76 snsspr1 4318 . . . . 5 {∅} ⊆ {∅, 2𝑜}
7775, 76jctil 559 . . . 4 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
7831, 36, 77rspcedvd 3305 . . 3 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
7923, 30, 78rspcedvd 3305 . 2 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ∃𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)))
809, 22, 79mp2an 707 1 𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 383  wa 384   = wceq 1480  wtru 1481  wcel 1987  wne 2790  wrex 2908  Vcvv 3189  wss 3559  c0 3896  ifcif 4063  𝒫 cpw 4135  {csn 4153  {cpr 4155  {ctp 4157  cmpt 4678  Oncon0 5687  suc csuc 5689  cfv 5852  1𝑜c1o 7505  2𝑜c2o 7506  3𝑜c3o 7507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909  ax-reg 8449
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-ord 5690  df-on 5691  df-suc 5693  df-iota 5815  df-fun 5854  df-fv 5860  df-1o 7512  df-2o 7513  df-3o 7514
This theorem is referenced by:  clsk1independent  37861
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