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Theorem clsk1indlem1 38660
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 6999 . . . . . 6 {∅, 1𝑜, 2𝑜} ∈ V
21a1i 11 . . . . 5 (⊤ → {∅, 1𝑜, 2𝑜} ∈ V)
3 snsstp1 4379 . . . . . 6 {∅} ⊆ {∅, 1𝑜, 2𝑜}
43a1i 11 . . . . 5 (⊤ → {∅} ⊆ {∅, 1𝑜, 2𝑜})
52, 4sselpwd 4840 . . . 4 (⊤ → {∅} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
65trud 1533 . . 3 {∅} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
7 df3o2 38639 . . . 4 3𝑜 = {∅, 1𝑜, 2𝑜}
87pweqi 4195 . . 3 𝒫 3𝑜 = 𝒫 {∅, 1𝑜, 2𝑜}
96, 8eleqtrri 2729 . 2 {∅} ∈ 𝒫 3𝑜
10 0ex 4823 . . . . . . . 8 ∅ ∈ V
1110snss 4348 . . . . . . 7 (∅ ∈ {∅, 1𝑜, 2𝑜} ↔ {∅} ⊆ {∅, 1𝑜, 2𝑜})
124, 11sylibr 224 . . . . . 6 (⊤ → ∅ ∈ {∅, 1𝑜, 2𝑜})
13 snsstp3 4381 . . . . . . . 8 {2𝑜} ⊆ {∅, 1𝑜, 2𝑜}
1413a1i 11 . . . . . . 7 (⊤ → {2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
15 2on 7613 . . . . . . . . 9 2𝑜 ∈ On
1615elexi 3244 . . . . . . . 8 2𝑜 ∈ V
1716snss 4348 . . . . . . 7 (2𝑜 ∈ {∅, 1𝑜, 2𝑜} ↔ {2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
1814, 17sylibr 224 . . . . . 6 (⊤ → 2𝑜 ∈ {∅, 1𝑜, 2𝑜})
1912, 18prssd 4386 . . . . 5 (⊤ → {∅, 2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
202, 19sselpwd 4840 . . . 4 (⊤ → {∅, 2𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
2120trud 1533 . . 3 {∅, 2𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
2221, 8eleqtrri 2729 . 2 {∅, 2𝑜} ∈ 𝒫 3𝑜
23 simpl 472 . . 3 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → {∅} ∈ 𝒫 3𝑜)
24 sseq1 3659 . . . . . 6 (𝑠 = {∅} → (𝑠𝑡 ↔ {∅} ⊆ 𝑡))
25 fveq2 6229 . . . . . . . 8 (𝑠 = {∅} → (𝐾𝑠) = (𝐾‘{∅}))
2625sseq1d 3665 . . . . . . 7 (𝑠 = {∅} → ((𝐾𝑠) ⊆ (𝐾𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
2726notbid 307 . . . . . 6 (𝑠 = {∅} → (¬ (𝐾𝑠) ⊆ (𝐾𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
2824, 27anbi12d 747 . . . . 5 (𝑠 = {∅} → ((𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
2928rexbidv 3081 . . . 4 (𝑠 = {∅} → (∃𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
3029adantl 481 . . 3 ((({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) ∧ 𝑠 = {∅}) → (∃𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
31 simpr 476 . . . 4 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → {∅, 2𝑜} ∈ 𝒫 3𝑜)
32 fveq2 6229 . . . . . . . 8 (𝑡 = {∅, 2𝑜} → (𝐾𝑡) = (𝐾‘{∅, 2𝑜}))
3332sseq2d 3666 . . . . . . 7 (𝑡 = {∅, 2𝑜} → ((𝐾‘{∅}) ⊆ (𝐾𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
3433notbid 307 . . . . . 6 (𝑡 = {∅, 2𝑜} → (¬ (𝐾‘{∅}) ⊆ (𝐾𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
3534cleq2lem 38231 . . . . 5 (𝑡 = {∅, 2𝑜} → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))))
3635adantl 481 . . . 4 ((({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) ∧ 𝑡 = {∅, 2𝑜}) → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))))
37 1on 7612 . . . . . . . . 9 1𝑜 ∈ On
3837elexi 3244 . . . . . . . 8 1𝑜 ∈ V
3938prid2 4330 . . . . . . 7 1𝑜 ∈ {∅, 1𝑜}
40 iftrue 4125 . . . . . . . . 9 (𝑟 = {∅} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = {∅, 1𝑜})
41 clsk1indlem.k . . . . . . . . 9 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
42 prex 4939 . . . . . . . . 9 {∅, 1𝑜} ∈ V
4340, 41, 42fvmpt 6321 . . . . . . . 8 ({∅} ∈ 𝒫 3𝑜 → (𝐾‘{∅}) = {∅, 1𝑜})
4443adantr 480 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → (𝐾‘{∅}) = {∅, 1𝑜})
4539, 44syl5eleqr 2737 . . . . . 6 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → 1𝑜 ∈ (𝐾‘{∅}))
46 1n0 7620 . . . . . . . . . . 11 1𝑜 ≠ ∅
4746neii 2825 . . . . . . . . . 10 ¬ 1𝑜 = ∅
48 eqcom 2658 . . . . . . . . . . . 12 (1𝑜 = 2𝑜 ↔ 2𝑜 = 1𝑜)
49 df-2o 7606 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
50 df-1o 7605 . . . . . . . . . . . . 13 1𝑜 = suc ∅
5149, 50eqeq12i 2665 . . . . . . . . . . . 12 (2𝑜 = 1𝑜 ↔ suc 1𝑜 = suc ∅)
52 suc11reg 8554 . . . . . . . . . . . 12 (suc 1𝑜 = suc ∅ ↔ 1𝑜 = ∅)
5348, 51, 523bitri 286 . . . . . . . . . . 11 (1𝑜 = 2𝑜 ↔ 1𝑜 = ∅)
5446, 53nemtbir 2918 . . . . . . . . . 10 ¬ 1𝑜 = 2𝑜
5547, 54pm3.2ni 917 . . . . . . . . 9 ¬ (1𝑜 = ∅ ∨ 1𝑜 = 2𝑜)
56 elpri 4230 . . . . . . . . 9 (1𝑜 ∈ {∅, 2𝑜} → (1𝑜 = ∅ ∨ 1𝑜 = 2𝑜))
5755, 56mto 188 . . . . . . . 8 ¬ 1𝑜 ∈ {∅, 2𝑜}
5857a1i 11 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ 1𝑜 ∈ {∅, 2𝑜})
59 eqeq1 2655 . . . . . . . . . . 11 (𝑟 = {∅, 2𝑜} → (𝑟 = {∅} ↔ {∅, 2𝑜} = {∅}))
60 id 22 . . . . . . . . . . 11 (𝑟 = {∅, 2𝑜} → 𝑟 = {∅, 2𝑜})
6159, 60ifbieq2d 4144 . . . . . . . . . 10 (𝑟 = {∅, 2𝑜} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if({∅, 2𝑜} = {∅}, {∅, 1𝑜}, {∅, 2𝑜}))
6216prid2 4330 . . . . . . . . . . . 12 2𝑜 ∈ {∅, 2𝑜}
63 2on0 7614 . . . . . . . . . . . . 13 2𝑜 ≠ ∅
64 nelsn 4245 . . . . . . . . . . . . 13 (2𝑜 ≠ ∅ → ¬ 2𝑜 ∈ {∅})
6563, 64ax-mp 5 . . . . . . . . . . . 12 ¬ 2𝑜 ∈ {∅}
66 nelneq2 2755 . . . . . . . . . . . 12 ((2𝑜 ∈ {∅, 2𝑜} ∧ ¬ 2𝑜 ∈ {∅}) → ¬ {∅, 2𝑜} = {∅})
6762, 65, 66mp2an 708 . . . . . . . . . . 11 ¬ {∅, 2𝑜} = {∅}
6867iffalsei 4129 . . . . . . . . . 10 if({∅, 2𝑜} = {∅}, {∅, 1𝑜}, {∅, 2𝑜}) = {∅, 2𝑜}
6961, 68syl6eq 2701 . . . . . . . . 9 (𝑟 = {∅, 2𝑜} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = {∅, 2𝑜})
70 prex 4939 . . . . . . . . 9 {∅, 2𝑜} ∈ V
7169, 41, 70fvmpt 6321 . . . . . . . 8 ({∅, 2𝑜} ∈ 𝒫 3𝑜 → (𝐾‘{∅, 2𝑜}) = {∅, 2𝑜})
7271adantl 481 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → (𝐾‘{∅, 2𝑜}) = {∅, 2𝑜})
7358, 72neleqtrrd 2752 . . . . . 6 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ 1𝑜 ∈ (𝐾‘{∅, 2𝑜}))
74 nelss 3697 . . . . . 6 ((1𝑜 ∈ (𝐾‘{∅}) ∧ ¬ 1𝑜 ∈ (𝐾‘{∅, 2𝑜})) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))
7545, 73, 74syl2anc 694 . . . . 5 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))
76 snsspr1 4377 . . . . 5 {∅} ⊆ {∅, 2𝑜}
7775, 76jctil 559 . . . 4 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
7831, 36, 77rspcedvd 3348 . . 3 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
7923, 30, 78rspcedvd 3348 . 2 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ∃𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)))
809, 22, 79mp2an 708 1 𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 382  wa 383   = wceq 1523  wtru 1524  wcel 2030  wne 2823  wrex 2942  Vcvv 3231  wss 3607  c0 3948  ifcif 4119  𝒫 cpw 4191  {csn 4210  {cpr 4212  {ctp 4214  cmpt 4762  Oncon0 5761  suc csuc 5763  cfv 5926  1𝑜c1o 7598  2𝑜c2o 7599  3𝑜c3o 7600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fv 5934  df-1o 7605  df-2o 7606  df-3o 7607
This theorem is referenced by:  clsk1independent  38661
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