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Theorem clsk1indlem1 37363
 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 6855 . . . . . 6 {∅, 1𝑜, 2𝑜} ∈ V
21a1i 11 . . . . 5 (⊤ → {∅, 1𝑜, 2𝑜} ∈ V)
3 snsstp1 4287 . . . . . 6 {∅} ⊆ {∅, 1𝑜, 2𝑜}
43a1i 11 . . . . 5 (⊤ → {∅} ⊆ {∅, 1𝑜, 2𝑜})
52, 4sselpwd 4734 . . . 4 (⊤ → {∅} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
65trud 1484 . . 3 {∅} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
7 df3o2 37342 . . . 4 3𝑜 = {∅, 1𝑜, 2𝑜}
87pweqi 4112 . . 3 𝒫 3𝑜 = 𝒫 {∅, 1𝑜, 2𝑜}
96, 8eleqtrri 2687 . 2 {∅} ∈ 𝒫 3𝑜
10 0ex 4718 . . . . . . . 8 ∅ ∈ V
1110snss 4259 . . . . . . 7 (∅ ∈ {∅, 1𝑜, 2𝑜} ↔ {∅} ⊆ {∅, 1𝑜, 2𝑜})
124, 11sylibr 223 . . . . . 6 (⊤ → ∅ ∈ {∅, 1𝑜, 2𝑜})
13 snsstp3 4289 . . . . . . . 8 {2𝑜} ⊆ {∅, 1𝑜, 2𝑜}
1413a1i 11 . . . . . . 7 (⊤ → {2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
15 2on 7455 . . . . . . . . 9 2𝑜 ∈ On
1615elexi 3186 . . . . . . . 8 2𝑜 ∈ V
1716snss 4259 . . . . . . 7 (2𝑜 ∈ {∅, 1𝑜, 2𝑜} ↔ {2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
1814, 17sylibr 223 . . . . . 6 (⊤ → 2𝑜 ∈ {∅, 1𝑜, 2𝑜})
1912, 18prssd 4294 . . . . 5 (⊤ → {∅, 2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
202, 19sselpwd 4734 . . . 4 (⊤ → {∅, 2𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
2120trud 1484 . . 3 {∅, 2𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
2221, 8eleqtrri 2687 . 2 {∅, 2𝑜} ∈ 𝒫 3𝑜
23 simpl 472 . . 3 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → {∅} ∈ 𝒫 3𝑜)
24 sseq1 3589 . . . . . 6 (𝑠 = {∅} → (𝑠𝑡 ↔ {∅} ⊆ 𝑡))
25 fveq2 6103 . . . . . . . 8 (𝑠 = {∅} → (𝐾𝑠) = (𝐾‘{∅}))
2625sseq1d 3595 . . . . . . 7 (𝑠 = {∅} → ((𝐾𝑠) ⊆ (𝐾𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
2726notbid 307 . . . . . 6 (𝑠 = {∅} → (¬ (𝐾𝑠) ⊆ (𝐾𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
2824, 27anbi12d 743 . . . . 5 (𝑠 = {∅} → ((𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)) ↔ ({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡))))
2928rexbidv 3034 . . . 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 6103 . . . . . . . 8 (𝑡 = {∅, 2𝑜} → (𝐾𝑡) = (𝐾‘{∅, 2𝑜}))
3332sseq2d 3596 . . . . . . 7 (𝑡 = {∅, 2𝑜} → ((𝐾‘{∅}) ⊆ (𝐾𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
3433notbid 307 . . . . . 6 (𝑡 = {∅, 2𝑜} → (¬ (𝐾‘{∅}) ⊆ (𝐾𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
3534cleq2lem 36933 . . . . 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 7454 . . . . . . . . 9 1𝑜 ∈ On
3837elexi 3186 . . . . . . . 8 1𝑜 ∈ V
3938prid2 4242 . . . . . . 7 1𝑜 ∈ {∅, 1𝑜}
40 iftrue 4042 . . . . . . . . 9 (𝑟 = {∅} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = {∅, 1𝑜})
41 clsk1indlem.k . . . . . . . . 9 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
42 prex 4836 . . . . . . . . 9 {∅, 1𝑜} ∈ V
4340, 41, 42fvmpt 6191 . . . . . . . 8 ({∅} ∈ 𝒫 3𝑜 → (𝐾‘{∅}) = {∅, 1𝑜})
4443adantr 480 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → (𝐾‘{∅}) = {∅, 1𝑜})
4539, 44syl5eleqr 2695 . . . . . 6 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → 1𝑜 ∈ (𝐾‘{∅}))
46 1n0 7462 . . . . . . . . . . 11 1𝑜 ≠ ∅
4746neii 2784 . . . . . . . . . 10 ¬ 1𝑜 = ∅
48 eqcom 2617 . . . . . . . . . . . 12 (1𝑜 = 2𝑜 ↔ 2𝑜 = 1𝑜)
49 df-2o 7448 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
50 df-1o 7447 . . . . . . . . . . . . 13 1𝑜 = suc ∅
5149, 50eqeq12i 2624 . . . . . . . . . . . 12 (2𝑜 = 1𝑜 ↔ suc 1𝑜 = suc ∅)
52 suc11reg 8399 . . . . . . . . . . . 12 (suc 1𝑜 = suc ∅ ↔ 1𝑜 = ∅)
5348, 51, 523bitri 285 . . . . . . . . . . 11 (1𝑜 = 2𝑜 ↔ 1𝑜 = ∅)
5446, 53nemtbir 2877 . . . . . . . . . 10 ¬ 1𝑜 = 2𝑜
5547, 54pm3.2ni 895 . . . . . . . . 9 ¬ (1𝑜 = ∅ ∨ 1𝑜 = 2𝑜)
56 elpri 4145 . . . . . . . . 9 (1𝑜 ∈ {∅, 2𝑜} → (1𝑜 = ∅ ∨ 1𝑜 = 2𝑜))
5755, 56mto 187 . . . . . . . 8 ¬ 1𝑜 ∈ {∅, 2𝑜}
5857a1i 11 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ 1𝑜 ∈ {∅, 2𝑜})
59 eqeq1 2614 . . . . . . . . . . 11 (𝑟 = {∅, 2𝑜} → (𝑟 = {∅} ↔ {∅, 2𝑜} = {∅}))
60 id 22 . . . . . . . . . . 11 (𝑟 = {∅, 2𝑜} → 𝑟 = {∅, 2𝑜})
6159, 60ifbieq2d 4061 . . . . . . . . . 10 (𝑟 = {∅, 2𝑜} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if({∅, 2𝑜} = {∅}, {∅, 1𝑜}, {∅, 2𝑜}))
6216prid2 4242 . . . . . . . . . . . 12 2𝑜 ∈ {∅, 2𝑜}
63 2on0 7456 . . . . . . . . . . . . 13 2𝑜 ≠ ∅
64 nelsn 4159 . . . . . . . . . . . . 13 (2𝑜 ≠ ∅ → ¬ 2𝑜 ∈ {∅})
6563, 64ax-mp 5 . . . . . . . . . . . 12 ¬ 2𝑜 ∈ {∅}
66 nelneq2 2713 . . . . . . . . . . . 12 ((2𝑜 ∈ {∅, 2𝑜} ∧ ¬ 2𝑜 ∈ {∅}) → ¬ {∅, 2𝑜} = {∅})
6762, 65, 66mp2an 704 . . . . . . . . . . 11 ¬ {∅, 2𝑜} = {∅}
6867iffalsei 4046 . . . . . . . . . 10 if({∅, 2𝑜} = {∅}, {∅, 1𝑜}, {∅, 2𝑜}) = {∅, 2𝑜}
6961, 68syl6eq 2660 . . . . . . . . 9 (𝑟 = {∅, 2𝑜} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = {∅, 2𝑜})
70 prex 4836 . . . . . . . . 9 {∅, 2𝑜} ∈ V
7169, 41, 70fvmpt 6191 . . . . . . . 8 ({∅, 2𝑜} ∈ 𝒫 3𝑜 → (𝐾‘{∅, 2𝑜}) = {∅, 2𝑜})
7271adantl 481 . . . . . . 7 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → (𝐾‘{∅, 2𝑜}) = {∅, 2𝑜})
7358, 72neleqtrrd 2710 . . . . . 6 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ 1𝑜 ∈ (𝐾‘{∅, 2𝑜}))
74 nelss 3627 . . . . . 6 ((1𝑜 ∈ (𝐾‘{∅}) ∧ ¬ 1𝑜 ∈ (𝐾‘{∅, 2𝑜})) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))
7545, 73, 74syl2anc 691 . . . . 5 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))
76 snsspr1 4285 . . . . 5 {∅} ⊆ {∅, 2𝑜}
7775, 76jctil 558 . . . 4 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
7831, 36, 77rspcedvd 3289 . . 3 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾𝑡)))
7923, 30, 78rspcedvd 3289 . 2 (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ∃𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡)))
809, 22, 79mp2an 704 1 𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ (𝐾𝑠) ⊆ (𝐾𝑡))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  {csn 4125  {cpr 4127  {ctp 4129   ↦ cmpt 4643  Oncon0 5640  suc csuc 5642  ‘cfv 5804  1𝑜c1o 7440  2𝑜c2o 7441  3𝑜c3o 7442 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-reg 8380 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fv 5812  df-1o 7447  df-2o 7448  df-3o 7449 This theorem is referenced by:  clsk1independent  37364
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