Theorem clsk1indlem1 38343

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 6957	. . . . . 6 ⊢ {∅, 1𝑜, 2𝑜} ∈ V
2	1	a1i 11	. . . . 5 ⊢ (⊤ → {∅, 1𝑜, 2𝑜} ∈ V)
3		snsstp1 4347	. . . . . 6 ⊢ {∅} ⊆ {∅, 1𝑜, 2𝑜}
4	3	a1i 11	. . . . 5 ⊢ (⊤ → {∅} ⊆ {∅, 1𝑜, 2𝑜})
5	2, 4	sselpwd 4807	. . . 4 ⊢ (⊤ → {∅} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
6	5	trud 1493	. . 3 ⊢ {∅} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
7		df3o2 38322	. . . 4 ⊢ 3𝑜 = {∅, 1𝑜, 2𝑜}
8	7	pweqi 4162	. . 3 ⊢ 𝒫 3𝑜 = 𝒫 {∅, 1𝑜, 2𝑜}
9	6, 8	eleqtrri 2700	. 2 ⊢ {∅} ∈ 𝒫 3𝑜
10		0ex 4790	. . . . . . . 8 ⊢ ∅ ∈ V
11	10	snss 4316	. . . . . . 7 ⊢ (∅ ∈ {∅, 1𝑜, 2𝑜} ↔ {∅} ⊆ {∅, 1𝑜, 2𝑜})
12	4, 11	sylibr 224	. . . . . 6 ⊢ (⊤ → ∅ ∈ {∅, 1𝑜, 2𝑜})
13		snsstp3 4349	. . . . . . . 8 ⊢ {2𝑜} ⊆ {∅, 1𝑜, 2𝑜}
14	13	a1i 11	. . . . . . 7 ⊢ (⊤ → {2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
15		2on 7568	. . . . . . . . 9 ⊢ 2𝑜 ∈ On
16	15	elexi 3213	. . . . . . . 8 ⊢ 2𝑜 ∈ V
17	16	snss 4316	. . . . . . 7 ⊢ (2𝑜 ∈ {∅, 1𝑜, 2𝑜} ↔ {2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
18	14, 17	sylibr 224	. . . . . 6 ⊢ (⊤ → 2𝑜 ∈ {∅, 1𝑜, 2𝑜})
19	12, 18	prssd 4354	. . . . 5 ⊢ (⊤ → {∅, 2𝑜} ⊆ {∅, 1𝑜, 2𝑜})
20	2, 19	sselpwd 4807	. . . 4 ⊢ (⊤ → {∅, 2𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜})
21	20	trud 1493	. . 3 ⊢ {∅, 2𝑜} ∈ 𝒫 {∅, 1𝑜, 2𝑜}
22	21, 8	eleqtrri 2700	. 2 ⊢ {∅, 2𝑜} ∈ 𝒫 3𝑜
23		simpl 473	. . 3 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → {∅} ∈ 𝒫 3𝑜)
24		sseq1 3626	. . . . . 6 ⊢ (𝑠 = {∅} → (𝑠 ⊆ 𝑡 ↔ {∅} ⊆ 𝑡))
25		fveq2 6191	. . . . . . . 8 ⊢ (𝑠 = {∅} → (𝐾‘𝑠) = (𝐾‘{∅}))
26	25	sseq1d 3632	. . . . . . 7 ⊢ (𝑠 = {∅} → ((𝐾‘𝑠) ⊆ (𝐾‘𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)))
27	26	notbid 308	. . . . . 6 ⊢ (𝑠 = {∅} → (¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)))
28	24, 27	anbi12d 747	. . . . 5 ⊢ (𝑠 = {∅} → ((𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) ↔ ({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡))))
29	28	rexbidv 3052	. . . 4 ⊢ (𝑠 = {∅} → (∃𝑡 ∈ 𝒫 3𝑜(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) ↔ ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡))))
30	29	adantl 482	. . 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 477	. . . 4 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → {∅, 2𝑜} ∈ 𝒫 3𝑜)
32		fveq2 6191	. . . . . . . 8 ⊢ (𝑡 = {∅, 2𝑜} → (𝐾‘𝑡) = (𝐾‘{∅, 2𝑜}))
33	32	sseq2d 3633	. . . . . . 7 ⊢ (𝑡 = {∅, 2𝑜} → ((𝐾‘{∅}) ⊆ (𝐾‘𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
34	33	notbid 308	. . . . . 6 ⊢ (𝑡 = {∅, 2𝑜} → (¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
35	34	cleq2lem 37914	. . . . 5 ⊢ (𝑡 = {∅, 2𝑜} → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)) ↔ ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))))
36	35	adantl 482	. . . 4 ⊢ ((({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) ∧ 𝑡 = {∅, 2𝑜}) → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)) ↔ ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))))
37		1on 7567	. . . . . . . . 9 ⊢ 1𝑜 ∈ On
38	37	elexi 3213	. . . . . . . 8 ⊢ 1𝑜 ∈ V
39	38	prid2 4298	. . . . . . 7 ⊢ 1𝑜 ∈ {∅, 1𝑜}
40		iftrue 4092	. . . . . . . . 9 ⊢ (𝑟 = {∅} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = {∅, 1𝑜})
41		clsk1indlem.k	. . . . . . . . 9 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
42		prex 4909	. . . . . . . . 9 ⊢ {∅, 1𝑜} ∈ V
43	40, 41, 42	fvmpt 6282	. . . . . . . 8 ⊢ ({∅} ∈ 𝒫 3𝑜 → (𝐾‘{∅}) = {∅, 1𝑜})
44	43	adantr 481	. . . . . . 7 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → (𝐾‘{∅}) = {∅, 1𝑜})
45	39, 44	syl5eleqr 2708	. . . . . 6 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → 1𝑜 ∈ (𝐾‘{∅}))
46		1n0 7575	. . . . . . . . . . 11 ⊢ 1𝑜 ≠ ∅
47	46	neii 2796	. . . . . . . . . 10 ⊢ ¬ 1𝑜 = ∅
48		eqcom 2629	. . . . . . . . . . . 12 ⊢ (1𝑜 = 2𝑜 ↔ 2𝑜 = 1𝑜)
49		df-2o 7561	. . . . . . . . . . . . 13 ⊢ 2𝑜 = suc 1𝑜
50		df-1o 7560	. . . . . . . . . . . . 13 ⊢ 1𝑜 = suc ∅
51	49, 50	eqeq12i 2636	. . . . . . . . . . . 12 ⊢ (2𝑜 = 1𝑜 ↔ suc 1𝑜 = suc ∅)
52		suc11reg 8516	. . . . . . . . . . . 12 ⊢ (suc 1𝑜 = suc ∅ ↔ 1𝑜 = ∅)
53	48, 51, 52	3bitri 286	. . . . . . . . . . 11 ⊢ (1𝑜 = 2𝑜 ↔ 1𝑜 = ∅)
54	46, 53	nemtbir 2889	. . . . . . . . . 10 ⊢ ¬ 1𝑜 = 2𝑜
55	47, 54	pm3.2ni 899	. . . . . . . . 9 ⊢ ¬ (1𝑜 = ∅ ∨ 1𝑜 = 2𝑜)
56		elpri 4197	. . . . . . . . 9 ⊢ (1𝑜 ∈ {∅, 2𝑜} → (1𝑜 = ∅ ∨ 1𝑜 = 2𝑜))
57	55, 56	mto 188	. . . . . . . 8 ⊢ ¬ 1𝑜 ∈ {∅, 2𝑜}
58	57	a1i 11	. . . . . . 7 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ 1𝑜 ∈ {∅, 2𝑜})
59		eqeq1 2626	. . . . . . . . . . 11 ⊢ (𝑟 = {∅, 2𝑜} → (𝑟 = {∅} ↔ {∅, 2𝑜} = {∅}))
60		id 22	. . . . . . . . . . 11 ⊢ (𝑟 = {∅, 2𝑜} → 𝑟 = {∅, 2𝑜})
61	59, 60	ifbieq2d 4111	. . . . . . . . . 10 ⊢ (𝑟 = {∅, 2𝑜} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = if({∅, 2𝑜} = {∅}, {∅, 1𝑜}, {∅, 2𝑜}))
62	16	prid2 4298	. . . . . . . . . . . 12 ⊢ 2𝑜 ∈ {∅, 2𝑜}
63		2on0 7569	. . . . . . . . . . . . 13 ⊢ 2𝑜 ≠ ∅
64		nelsn 4212	. . . . . . . . . . . . 13 ⊢ (2𝑜 ≠ ∅ → ¬ 2𝑜 ∈ {∅})
65	63, 64	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 2𝑜 ∈ {∅}
66		nelneq2 2726	. . . . . . . . . . . 12 ⊢ ((2𝑜 ∈ {∅, 2𝑜} ∧ ¬ 2𝑜 ∈ {∅}) → ¬ {∅, 2𝑜} = {∅})
67	62, 65, 66	mp2an 708	. . . . . . . . . . 11 ⊢ ¬ {∅, 2𝑜} = {∅}
68	67	iffalsei 4096	. . . . . . . . . 10 ⊢ if({∅, 2𝑜} = {∅}, {∅, 1𝑜}, {∅, 2𝑜}) = {∅, 2𝑜}
69	61, 68	syl6eq 2672	. . . . . . . . 9 ⊢ (𝑟 = {∅, 2𝑜} → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) = {∅, 2𝑜})
70		prex 4909	. . . . . . . . 9 ⊢ {∅, 2𝑜} ∈ V
71	69, 41, 70	fvmpt 6282	. . . . . . . 8 ⊢ ({∅, 2𝑜} ∈ 𝒫 3𝑜 → (𝐾‘{∅, 2𝑜}) = {∅, 2𝑜})
72	71	adantl 482	. . . . . . 7 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → (𝐾‘{∅, 2𝑜}) = {∅, 2𝑜})
73	58, 72	neleqtrrd 2723	. . . . . 6 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ 1𝑜 ∈ (𝐾‘{∅, 2𝑜}))
74		nelss 3664	. . . . . 6 ⊢ ((1𝑜 ∈ (𝐾‘{∅}) ∧ ¬ 1𝑜 ∈ (𝐾‘{∅, 2𝑜})) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))
75	45, 73, 74	syl2anc 693	. . . . 5 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜}))
76		snsspr1 4345	. . . . 5 ⊢ {∅} ⊆ {∅, 2𝑜}
77	75, 76	jctil 560	. . . 4 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ({∅} ⊆ {∅, 2𝑜} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2𝑜})))
78	31, 36, 77	rspcedvd 3317	. . 3 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ∃𝑡 ∈ 𝒫 3𝑜({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)))
79	23, 30, 78	rspcedvd 3317	. 2 ⊢ (({∅} ∈ 𝒫 3𝑜 ∧ {∅, 2𝑜} ∈ 𝒫 3𝑜) → ∃𝑠 ∈ 𝒫 3𝑜∃𝑡 ∈ 𝒫 3𝑜(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)))
80	9, 22, 79	mp2an 708	1 ⊢ ∃𝑠 ∈ 𝒫 3𝑜∃𝑡 ∈ 𝒫 3𝑜(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡))

Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483  ⊤wtru 1484   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  {csn 4177  {cpr 4179  {ctp 4181   ↦ cmpt 4729  Oncon0 5723  suc csuc 5725  ‘cfv 5888  1_𝑜c1o 7553  2_𝑜c2o 7554  3_𝑜c3o 7555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-1o 7560  df-2o 7561  df-3o 7562

This theorem is referenced by:  clsk1independent  38344