Theorem clsk1indlem1 37363

Description: The ansatz closure function (𝑟 ∈ 𝒫 3_𝑜 ↦ if(𝑟 = {∅}, {∅, 1_𝑜}, 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 3𝑜∃𝑡 ∈ 𝒫 3𝑜(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡))

Distinct variable groups:   𝐾,𝑠,𝑡   𝑠,𝑟,𝑡

Allowed substitution hint:   𝐾(𝑟)

Proof of Theorem clsk1indlem1

Step	Hyp	Ref	Expression
1		tpex 6855	. . . . . 6 ⊢ {∅, 1𝑜, 2𝑜} ∈ V
2	1	a1i 11	. . . . 5 ⊢ (⊤ → {∅, 1𝑜, 2𝑜} ∈ V)
3		snsstp1 4287	. . . . . 6 ⊢ {∅} ⊆ {∅, 1𝑜, 2𝑜}
4	3	a1i 11	. . . . 5 ⊢ (⊤ → {∅} ⊆ {∅, 1𝑜, 2𝑜})
5	2, 4	sselpwd 4734	. . . 4 ⊢ (⊤ → {∅} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
6	5	trud 1484	. . 3 ⊢ {∅} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
7		df3o2 37342	. . . 4 ⊢ 3𝑜 = {∅, 1𝑜, 2𝑜}
8	7	pweqi 4112	. . 3 ⊢ 𝒫 3𝑜 = 𝒫 {∅, 1𝑜, 2𝑜}
9	6, 8	eleqtrri 2687	. 2 ⊢ {∅} ∈ 𝒫 3𝑜
10		0ex 4718	. . . . . . . 8 ⊢ ∅ ∈ V
11	10	snss 4259	. . . . . . 7 ⊢ (∅ ∈ {∅, 1𝑜, 2𝑜} ↔ {∅} ⊆ {∅, 1𝑜, 2𝑜})
12	4, 11	sylibr 223	. . . . . 6 ⊢ (⊤ → ∅ ∈ {∅, 1𝑜, 2𝑜})
13		snsstp3 4289	. . . . . . . 8 ⊢ {2𝑜} ⊆ {∅, 1𝑜, 2𝑜}
14	13	a1i 11	. . . . . . 7 ⊢ (⊤ → {2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
15		2on 7455	. . . . . . . . 9 ⊢ 2𝑜 ∈ On
16	15	elexi 3186	. . . . . . . 8 ⊢ 2𝑜 ∈ V
17	16	snss 4259	. . . . . . 7 ⊢ (2𝑜 ∈ {∅, 1𝑜, 2𝑜} ↔ {2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
18	14, 17	sylibr 223	. . . . . 6 ⊢ (⊤ → 2𝑜 ∈ {∅, 1𝑜, 2𝑜})
19	12, 18	prssd 4294	. . . . 5 ⊢ (⊤ → {∅, 2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
20	2, 19	sselpwd 4734	. . . 4 ⊢ (⊤ → {∅, 2𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
21	20	trud 1484	. . 3 ⊢ {∅, 2𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
22	21, 8	eleqtrri 2687	. 2 ⊢ {∅, 2𝑜} ∈ 𝒫 3𝑜
23		simpl 472	. . 3 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → {∅} ∈ 𝒫 3𝑜)
24		sseq1 3589	. . . . . 6 ⊢ (𝑠 = {∅} → (𝑠 ⊆ 𝑡 ↔ {∅} ⊆ 𝑡))
25		fveq2 6103	. . . . . . . 8 ⊢ (𝑠 = {∅} → (𝐾‘𝑠) = (𝐾‘{∅}))
26	25	sseq1d 3595	. . . . . . 7 ⊢ (𝑠 = {∅} → ((𝐾‘𝑠) ⊆ (𝐾‘𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)))
27	26	notbid 307	. . . . . 6 ⊢ (𝑠 = {∅} → (¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)))
28	24, 27	anbi12d 743	. . . . 5 ⊢ (𝑠 = {∅} → ((𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) ↔ ({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡))))
29	28	rexbidv 3034	. . . 4 ⊢ (𝑠 = {∅} → (∃𝑡 ∈ 𝒫 3𝑜(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) ↔ ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡))))
30	29	adantl 481	. . 3 ⊢ ((({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) ∧ 𝑠 = {∅}) → (∃𝑡 ∈ 𝒫 3𝑜(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) ↔ ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡))))
31		simpr 476	. . . 4 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → {∅, 2𝑜} ∈ 𝒫 3𝑜)
32		fveq2 6103	. . . . . . . 8 ⊢ (𝑡 = {∅, 2𝑜} → (𝐾‘𝑡) = (𝐾‘{∅, 2𝑜}))
33	32	sseq2d 3596	. . . . . . 7 ⊢ (𝑡 = {∅, 2𝑜} → ((𝐾‘{∅}) ⊆ (𝐾‘𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
34	33	notbid 307	. . . . . 6 ⊢ (𝑡 = {∅, 2𝑜} → (¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
35	34	cleq2lem 36933	. . . . 5 ⊢ (𝑡 = {∅, 2𝑜} → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)) ↔ ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))))
36	35	adantl 481	. . . 4 ⊢ ((({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) ∧ 𝑡 = {∅, 2𝑜}) → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)) ↔ ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))))
37		1on 7454	. . . . . . . . 9 ⊢ 1𝑜 ∈ On
38	37	elexi 3186	. . . . . . . 8 ⊢ 1𝑜 ∈ V
39	38	prid2 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
43	40, 41, 42	fvmpt 6191	. . . . . . . 8 ⊢ ({∅} ∈ 𝒫 3𝑜 → (𝐾‘{∅}) = {∅, 1𝑜})
44	43	adantr 480	. . . . . . 7 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → (𝐾‘{∅}) = {∅, 1𝑜})
45	39, 44	syl5eleqr 2695	. . . . . 6 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → 1𝑜 ∈ (𝐾‘{∅}))
46		1n0 7462	. . . . . . . . . . 11 ⊢ 1𝑜 ≠ ∅
47	46	neii 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 ∅
51	49, 50	eqeq12i 2624	. . . . . . . . . . . 12 ⊢ (2𝑜 = 1𝑜 ↔ suc 1𝑜 = suc ∅)
52		suc11reg 8399	. . . . . . . . . . . 12 ⊢ (suc 1𝑜 = suc ∅ ↔ 1𝑜 = ∅)
53	48, 51, 52	3bitri 285	. . . . . . . . . . 11 ⊢ (1𝑜 = 2𝑜 ↔ 1𝑜 = ∅)
54	46, 53	nemtbir 2877	. . . . . . . . . 10 ⊢ ¬ 1𝑜 = 2𝑜
55	47, 54	pm3.2ni 895	. . . . . . . . 9 ⊢ ¬ (1𝑜 = ∅ ∨ 1𝑜 = 2𝑜)
56		elpri 4145	. . . . . . . . 9 ⊢ (1𝑜 ∈ {∅, 2𝑜} → (1𝑜 = ∅ ∨ 1𝑜 = 2𝑜))
57	55, 56	mto 187	. . . . . . . 8 ⊢ ¬ 1𝑜 ∈ {∅, 2𝑜}
58	57	a1i 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})
61	59, 60	ifbieq2d 4061	. . . . . . . . . 10 ⊢ (𝑟 = {∅, 2𝑜} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if({∅, 2𝑜} = {∅}, {∅, 1𝑜}, {∅, 2𝑜}))
62	16	prid2 4242	. . . . . . . . . . . 12 ⊢ 2𝑜 ∈ {∅, 2𝑜}
63		2on0 7456	. . . . . . . . . . . . 13 ⊢ 2𝑜 ≠ ∅
64		nelsn 4159	. . . . . . . . . . . . 13 ⊢ (2𝑜 ≠ ∅ → ¬ 2𝑜 ∈ {∅})
65	63, 64	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 2𝑜 ∈ {∅}
66		nelneq2 2713	. . . . . . . . . . . 12 ⊢ ((2𝑜 ∈ {∅, 2𝑜} ∧ ¬ 2𝑜 ∈ {∅}) → ¬ {∅, 2𝑜} = {∅})
67	62, 65, 66	mp2an 704	. . . . . . . . . . 11 ⊢ ¬ {∅, 2𝑜} = {∅}
68	67	iffalsei 4046	. . . . . . . . . 10 ⊢ if({∅, 2𝑜} = {∅}, {∅, 1𝑜}, {∅, 2𝑜}) = {∅, 2𝑜}
69	61, 68	syl6eq 2660	. . . . . . . . 9 ⊢ (𝑟 = {∅, 2𝑜} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = {∅, 2𝑜})
70		prex 4836	. . . . . . . . 9 ⊢ {∅, 2𝑜} ∈ V
71	69, 41, 70	fvmpt 6191	. . . . . . . 8 ⊢ ({∅, 2𝑜} ∈ 𝒫 3𝑜 → (𝐾‘{∅, 2𝑜}) = {∅, 2𝑜})
72	71	adantl 481	. . . . . . 7 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → (𝐾‘{∅, 2𝑜}) = {∅, 2𝑜})
73	58, 72	neleqtrrd 2710	. . . . . 6 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ 1𝑜 ∈ (𝐾‘{∅, 2𝑜}))
74		nelss 3627	. . . . . 6 ⊢ ((1𝑜 ∈ (𝐾‘{∅}) ∧ ¬ 1𝑜 ∈ (𝐾‘{∅, 2𝑜})) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))
75	45, 73, 74	syl2anc 691	. . . . 5 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))
76		snsspr1 4285	. . . . 5 ⊢ {∅} ⊆ {∅, 2𝑜}
77	75, 76	jctil 558	. . . 4 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
78	31, 36, 77	rspcedvd 3289	. . 3 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)))
79	23, 30, 78	rspcedvd 3289	. 2 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ∃𝑠 ∈ 𝒫 3𝑜∃𝑡 ∈ 𝒫 3𝑜(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)))
80	9, 22, 79	mp2an 704	1 ⊢ ∃𝑠 ∈ 𝒫 3𝑜∃𝑡 ∈ 𝒫 3𝑜(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡))

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  1_𝑜c1o 7440  2_𝑜c2o 7441  3_𝑜c3o 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