Theorem clsk1indlem1 44791

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 7743	. . . 4 ⊢ {∅, 1o, 2o} ∈ V
2		snsstp1 4782	. . . 4 ⊢ {∅} ⊆ {∅, 1o, 2o}
3	1, 2	elpwi2 5306	. . 3 ⊢ {∅} ∈ 𝒫 {∅, 1o, 2o}
4		df3o2 44060	. . . 4 ⊢ 3o = {∅, 1o, 2o}
5	4	pweqi 4578	. . 3 ⊢ 𝒫 3o = 𝒫 {∅, 1o, 2o}
6	3, 5	eleqtrri 2862	. 2 ⊢ {∅} ∈ 𝒫 3o
7	1	a1i 11	. . . . 5 ⊢ (⊤ → {∅, 1o, 2o} ∈ V)
8	2	a1i 11	. . . . . . 7 ⊢ (⊤ → {∅} ⊆ {∅, 1o, 2o})
9		0ex 5270	. . . . . . . 8 ⊢ ∅ ∈ V
10	9	snss 4750	. . . . . . 7 ⊢ (∅ ∈ {∅, 1o, 2o} ↔ {∅} ⊆ {∅, 1o, 2o})
11	8, 10	sylibr 237	. . . . . 6 ⊢ (⊤ → ∅ ∈ {∅, 1o, 2o})
12		snsstp3 4784	. . . . . . . 8 ⊢ {2o} ⊆ {∅, 1o, 2o}
13	12	a1i 11	. . . . . . 7 ⊢ (⊤ → {2o} ⊆ {∅, 1o, 2o})
14		2oex 8461	. . . . . . . 8 ⊢ 2o ∈ V
15	14	snss 4750	. . . . . . 7 ⊢ (2o ∈ {∅, 1o, 2o} ↔ {2o} ⊆ {∅, 1o, 2o})
16	13, 15	sylibr 237	. . . . . 6 ⊢ (⊤ → 2o ∈ {∅, 1o, 2o})
17	11, 16	prssd 4788	. . . . 5 ⊢ (⊤ → {∅, 2o} ⊆ {∅, 1o, 2o})
18	7, 17	sselpwd 5299	. . . 4 ⊢ (⊤ → {∅, 2o} ∈ 𝒫 {∅, 1o, 2o})
19	18	mptru 1577	. . 3 ⊢ {∅, 2o} ∈ 𝒫 {∅, 1o, 2o}
20	19, 5	eleqtrri 2862	. 2 ⊢ {∅, 2o} ∈ 𝒫 3o
21		simpl 487	. . 3 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → {∅} ∈ 𝒫 3o)
22		sseq1 3962	. . . . . 6 ⊢ (𝑠 = {∅} → (𝑠 ⊆ 𝑡 ↔ {∅} ⊆ 𝑡))
23		fveq2 6881	. . . . . . . 8 ⊢ (𝑠 = {∅} → (𝐾‘𝑠) = (𝐾‘{∅}))
24	23	sseq1d 3968	. . . . . . 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 3189	. . . 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 6881	. . . . . . . 8 ⊢ (𝑡 = {∅, 2o} → (𝐾‘𝑡) = (𝐾‘{∅, 2o}))
31	30	sseq2d 3969	. . . . . . 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 44354	. . . . 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		1oelpr 8460	. . . . . . 7 ⊢ 1o ∈ {∅, 1o}
36		iftrue 4493	. . . . . . . . 9 ⊢ (𝑟 = {∅} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = {∅, 1o})
37		clsk1indlem.k	. . . . . . . . 9 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
38		prex 5409	. . . . . . . . 9 ⊢ {∅, 1o} ∈ V
39	36, 37, 38	fvmpt 6989	. . . . . . . 8 ⊢ ({∅} ∈ 𝒫 3o → (𝐾‘{∅}) = {∅, 1o})
40	39	adantr 485	. . . . . . 7 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → (𝐾‘{∅}) = {∅, 1o})
41	35, 40	eleqtrrid 2870	. . . . . 6 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → 1o ∈ (𝐾‘{∅}))
42		1n0 8468	. . . . . . . . . . 11 ⊢ 1o ≠ ∅
43	42	neii 2960	. . . . . . . . . 10 ⊢ ¬ 1o = ∅
44		eqcom 2770	. . . . . . . . . . . 12 ⊢ (1o = 2o ↔ 2o = 1o)
45		df-2o 8450	. . . . . . . . . . . . 13 ⊢ 2o = suc 1o
46		df-1o 8449	. . . . . . . . . . . . 13 ⊢ 1o = suc ∅
47	45, 46	eqeq12i 2781	. . . . . . . . . . . 12 ⊢ (2o = 1o ↔ suc 1o = suc ∅)
48		suc11reg 9584	. . . . . . . . . . . 12 ⊢ (suc 1o = suc ∅ ↔ 1o = ∅)
49	44, 47, 48	3bitri 300	. . . . . . . . . . 11 ⊢ (1o = 2o ↔ 1o = ∅)
50	42, 49	nemtbir 3054	. . . . . . . . . 10 ⊢ ¬ 1o = 2o
51	43, 50	pm3.2ni 893	. . . . . . . . 9 ⊢ ¬ (1o = ∅ ∨ 1o = 2o)
52		elpri 4613	. . . . . . . . 9 ⊢ (1o ∈ {∅, 2o} → (1o = ∅ ∨ 1o = 2o))
53	51, 52	mto 200	. . . . . . . 8 ⊢ ¬ 1o ∈ {∅, 2o}
54	53	a1i 11	. . . . . . 7 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ 1o ∈ {∅, 2o})
55		eqeq1 2767	. . . . . . . . . . 11 ⊢ (𝑟 = {∅, 2o} → (𝑟 = {∅} ↔ {∅, 2o} = {∅}))
56		id 23	. . . . . . . . . . 11 ⊢ (𝑟 = {∅, 2o} → 𝑟 = {∅, 2o})
57	55, 56	ifbieq2d 4514	. . . . . . . . . 10 ⊢ (𝑟 = {∅, 2o} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if({∅, 2o} = {∅}, {∅, 1o}, {∅, 2o}))
58	14	prid2 4729	. . . . . . . . . . . 12 ⊢ 2o ∈ {∅, 2o}
59		2on0 8464	. . . . . . . . . . . . 13 ⊢ 2o ≠ ∅
60		nelsn 4632	. . . . . . . . . . . . 13 ⊢ (2o ≠ ∅ → ¬ 2o ∈ {∅})
61	59, 60	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 2o ∈ {∅}
62		nelneq2 2888	. . . . . . . . . . . 12 ⊢ ((2o ∈ {∅, 2o} ∧ ¬ 2o ∈ {∅}) → ¬ {∅, 2o} = {∅})
63	58, 61, 62	mp2an 704	. . . . . . . . . . 11 ⊢ ¬ {∅, 2o} = {∅}
64	63	iffalsei 4497	. . . . . . . . . 10 ⊢ if({∅, 2o} = {∅}, {∅, 1o}, {∅, 2o}) = {∅, 2o}
65	57, 64	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑟 = {∅, 2o} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = {∅, 2o})
66		prex 5409	. . . . . . . . 9 ⊢ {∅, 2o} ∈ V
67	65, 37, 66	fvmpt 6989	. . . . . . . 8 ⊢ ({∅, 2o} ∈ 𝒫 3o → (𝐾‘{∅, 2o}) = {∅, 2o})
68	67	adantl 486	. . . . . . 7 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → (𝐾‘{∅, 2o}) = {∅, 2o})
69	54, 68	neleqtrrd 2886	. . . . . 6 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ 1o ∈ (𝐾‘{∅, 2o}))
70		nelss 4003	. . . . . 6 ⊢ ((1o ∈ (𝐾‘{∅}) ∧ ¬ 1o ∈ (𝐾‘{∅, 2o})) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))
71	41, 69, 70	syl2anc 595	. . . . 5 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))
72		snsspr1 4780	. . . . 5 ⊢ {∅} ⊆ {∅, 2o}
73	71, 72	jctil 528	. . . 4 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ({∅} ⊆ {∅, 2o} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o})))
74	29, 34, 73	rspcedvd 3583	. . 3 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ∃𝑡 ∈ 𝒫 3o({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)))
75	21, 28, 74	rspcedvd 3583	. 2 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ∃𝑠 ∈ 𝒫 3o∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)))
76	6, 20, 75	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 1570  ⊤wtru 1571   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  ifcif 4487  𝒫 cpw 4562  {csn 4589  {cpr 4591  {ctp 4593   ↦ cmpt 5192  suc csuc 6362  ‘cfv 6536  1_oc1o 8442  2_oc2o 8443  3_oc3o 8444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732  ax-reg 9550

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-eprel 5561  df-fr 5614  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-suc 6366  df-iota 6492  df-fun 6538  df-fv 6544  df-1o 8449  df-2o 8450  df-3o 8451

This theorem is referenced by:  clsk1independent  44792