Theorem clsk1indlem1 44697

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 7745	. . . 4 ⊢ {∅, 1o, 2o} ∈ V
2		snsstp1 4786	. . . 4 ⊢ {∅} ⊆ {∅, 1o, 2o}
3	1, 2	elpwi2 5306	. . 3 ⊢ {∅} ∈ 𝒫 {∅, 1o, 2o}
4		df3o2 43966	. . . 4 ⊢ 3o = {∅, 1o, 2o}
5	4	pweqi 4583	. . 3 ⊢ 𝒫 3o = 𝒫 {∅, 1o, 2o}
6	3, 5	eleqtrri 2868	. 2 ⊢ {∅} ∈ 𝒫 3o
7	1	a1i 11	. . . . 5 ⊢ (⊤ → {∅, 1o, 2o} ∈ V)
8	2	a1i 11	. . . . . . 7 ⊢ (⊤ → {∅} ⊆ {∅, 1o, 2o})
9		0ex 5272	. . . . . . . 8 ⊢ ∅ ∈ V
10	9	snss 4755	. . . . . . 7 ⊢ (∅ ∈ {∅, 1o, 2o} ↔ {∅} ⊆ {∅, 1o, 2o})
11	8, 10	sylibr 237	. . . . . 6 ⊢ (⊤ → ∅ ∈ {∅, 1o, 2o})
12		snsstp3 4788	. . . . . . . 8 ⊢ {2o} ⊆ {∅, 1o, 2o}
13	12	a1i 11	. . . . . . 7 ⊢ (⊤ → {2o} ⊆ {∅, 1o, 2o})
14		2oex 8465	. . . . . . . 8 ⊢ 2o ∈ V
15	14	snss 4755	. . . . . . 7 ⊢ (2o ∈ {∅, 1o, 2o} ↔ {2o} ⊆ {∅, 1o, 2o})
16	13, 15	sylibr 237	. . . . . 6 ⊢ (⊤ → 2o ∈ {∅, 1o, 2o})
17	11, 16	prssd 4792	. . . . 5 ⊢ (⊤ → {∅, 2o} ⊆ {∅, 1o, 2o})
18	7, 17	sselpwd 5299	. . . 4 ⊢ (⊤ → {∅, 2o} ∈ 𝒫 {∅, 1o, 2o})
19	18	mptru 1574	. . 3 ⊢ {∅, 2o} ∈ 𝒫 {∅, 1o, 2o}
20	19, 5	eleqtrri 2868	. 2 ⊢ {∅, 2o} ∈ 𝒫 3o
21		simpl 487	. . 3 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → {∅} ∈ 𝒫 3o)
22		sseq1 3970	. . . . . 6 ⊢ (𝑠 = {∅} → (𝑠 ⊆ 𝑡 ↔ {∅} ⊆ 𝑡))
23		fveq2 6882	. . . . . . . 8 ⊢ (𝑠 = {∅} → (𝐾‘𝑠) = (𝐾‘{∅}))
24	23	sseq1d 3976	. . . . . . 7 ⊢ (𝑠 = {∅} → ((𝐾‘𝑠) ⊆ (𝐾‘𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)))
25	24	notbid 321	. . . . . 6 ⊢ (𝑠 = {∅} → (¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)))
26	22, 25	anbi12d 643	. . . . 5 ⊢ (𝑠 = {∅} → ((𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) ↔ ({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡))))
27	26	rexbidv 3195	. . . 4 ⊢ (𝑠 = {∅} → (∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) ↔ ∃𝑡 ∈ 𝒫 3o({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡))))
28	27	adantl 486	. . 3 ⊢ ((({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) ∧ 𝑠 = {∅}) → (∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) ↔ ∃𝑡 ∈ 𝒫 3o({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡))))
29		simpr 489	. . . 4 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → {∅, 2o} ∈ 𝒫 3o)
30		fveq2 6882	. . . . . . . 8 ⊢ (𝑡 = {∅, 2o} → (𝐾‘𝑡) = (𝐾‘{∅, 2o}))
31	30	sseq2d 3977	. . . . . . 7 ⊢ (𝑡 = {∅, 2o} → ((𝐾‘{∅}) ⊆ (𝐾‘𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o})))
32	31	notbid 321	. . . . . 6 ⊢ (𝑡 = {∅, 2o} → (¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o})))
33	32	cleq2lem 44260	. . . . 5 ⊢ (𝑡 = {∅, 2o} → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)) ↔ ({∅} ⊆ {∅, 2o} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))))
34	33	adantl 486	. . . 4 ⊢ ((({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) ∧ 𝑡 = {∅, 2o}) → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)) ↔ ({∅} ⊆ {∅, 2o} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))))
35		1oex 8463	. . . . . . . 8 ⊢ 1o ∈ V
36	35	prid2 4734	. . . . . . 7 ⊢ 1o ∈ {∅, 1o}
37		iftrue 4498	. . . . . . . . 9 ⊢ (𝑟 = {∅} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = {∅, 1o})
38		clsk1indlem.k	. . . . . . . . 9 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
39		prex 5410	. . . . . . . . 9 ⊢ {∅, 1o} ∈ V
40	37, 38, 39	fvmpt 6990	. . . . . . . 8 ⊢ ({∅} ∈ 𝒫 3o → (𝐾‘{∅}) = {∅, 1o})
41	40	adantr 485	. . . . . . 7 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → (𝐾‘{∅}) = {∅, 1o})
42	36, 41	eleqtrrid 2876	. . . . . 6 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → 1o ∈ (𝐾‘{∅}))
43		1n0 8472	. . . . . . . . . . 11 ⊢ 1o ≠ ∅
44	43	neii 2966	. . . . . . . . . 10 ⊢ ¬ 1o = ∅
45		eqcom 2776	. . . . . . . . . . . 12 ⊢ (1o = 2o ↔ 2o = 1o)
46		df-2o 8454	. . . . . . . . . . . . 13 ⊢ 2o = suc 1o
47		df-1o 8453	. . . . . . . . . . . . 13 ⊢ 1o = suc ∅
48	46, 47	eqeq12i 2787	. . . . . . . . . . . 12 ⊢ (2o = 1o ↔ suc 1o = suc ∅)
49		suc11reg 9588	. . . . . . . . . . . 12 ⊢ (suc 1o = suc ∅ ↔ 1o = ∅)
50	45, 48, 49	3bitri 300	. . . . . . . . . . 11 ⊢ (1o = 2o ↔ 1o = ∅)
51	43, 50	nemtbir 3060	. . . . . . . . . 10 ⊢ ¬ 1o = 2o
52	44, 51	pm3.2ni 893	. . . . . . . . 9 ⊢ ¬ (1o = ∅ ∨ 1o = 2o)
53		elpri 4618	. . . . . . . . 9 ⊢ (1o ∈ {∅, 2o} → (1o = ∅ ∨ 1o = 2o))
54	52, 53	mto 200	. . . . . . . 8 ⊢ ¬ 1o ∈ {∅, 2o}
55	54	a1i 11	. . . . . . 7 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ 1o ∈ {∅, 2o})
56		eqeq1 2773	. . . . . . . . . . 11 ⊢ (𝑟 = {∅, 2o} → (𝑟 = {∅} ↔ {∅, 2o} = {∅}))
57		id 23	. . . . . . . . . . 11 ⊢ (𝑟 = {∅, 2o} → 𝑟 = {∅, 2o})
58	56, 57	ifbieq2d 4519	. . . . . . . . . 10 ⊢ (𝑟 = {∅, 2o} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if({∅, 2o} = {∅}, {∅, 1o}, {∅, 2o}))
59	14	prid2 4734	. . . . . . . . . . . 12 ⊢ 2o ∈ {∅, 2o}
60		2on0 8468	. . . . . . . . . . . . 13 ⊢ 2o ≠ ∅
61		nelsn 4637	. . . . . . . . . . . . 13 ⊢ (2o ≠ ∅ → ¬ 2o ∈ {∅})
62	60, 61	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 2o ∈ {∅}
63		nelneq2 2894	. . . . . . . . . . . 12 ⊢ ((2o ∈ {∅, 2o} ∧ ¬ 2o ∈ {∅}) → ¬ {∅, 2o} = {∅})
64	59, 62, 63	mp2an 704	. . . . . . . . . . 11 ⊢ ¬ {∅, 2o} = {∅}
65	64	iffalsei 4502	. . . . . . . . . 10 ⊢ if({∅, 2o} = {∅}, {∅, 1o}, {∅, 2o}) = {∅, 2o}
66	58, 65	eqtrdi 2820	. . . . . . . . 9 ⊢ (𝑟 = {∅, 2o} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = {∅, 2o})
67		prex 5410	. . . . . . . . 9 ⊢ {∅, 2o} ∈ V
68	66, 38, 67	fvmpt 6990	. . . . . . . 8 ⊢ ({∅, 2o} ∈ 𝒫 3o → (𝐾‘{∅, 2o}) = {∅, 2o})
69	68	adantl 486	. . . . . . 7 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → (𝐾‘{∅, 2o}) = {∅, 2o})
70	55, 69	neleqtrrd 2892	. . . . . 6 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ 1o ∈ (𝐾‘{∅, 2o}))
71		nelss 4011	. . . . . 6 ⊢ ((1o ∈ (𝐾‘{∅}) ∧ ¬ 1o ∈ (𝐾‘{∅, 2o})) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))
72	42, 70, 71	syl2anc 595	. . . . 5 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))
73		snsspr1 4784	. . . . 5 ⊢ {∅} ⊆ {∅, 2o}
74	72, 73	jctil 528	. . . 4 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ({∅} ⊆ {∅, 2o} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o})))
75	29, 34, 74	rspcedvd 3592	. . 3 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ∃𝑡 ∈ 𝒫 3o({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)))
76	21, 28, 75	rspcedvd 3592	. 2 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ∃𝑠 ∈ 𝒫 3o∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)))
77	6, 20, 76	mp2an 704	1 ⊢ ∃𝑠 ∈ 𝒫 3o∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567  ⊤wtru 1568   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  ifcif 4492  𝒫 cpw 4567  {csn 4594  {cpr 4596  {ctp 4598   ↦ cmpt 5196  suc csuc 6363  ‘cfv 6537  1_oc1o 8446  2_oc2o 8447  3_oc3o 8448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733  ax-reg 9554

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-eprel 5562  df-fr 5615  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-suc 6367  df-iota 6493  df-fun 6539  df-fv 6545  df-1o 8453  df-2o 8454  df-3o 8455

This theorem is referenced by:  clsk1independent  44698