Theorem clsk1indlem1 41662

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 7606	. . . 4 ⊢ {∅, 1o, 2o} ∈ V
2		snsstp1 4750	. . . 4 ⊢ {∅} ⊆ {∅, 1o, 2o}
3	1, 2	elpwi2 5271	. . 3 ⊢ {∅} ∈ 𝒫 {∅, 1o, 2o}
4		df3o2 41641	. . . 4 ⊢ 3o = {∅, 1o, 2o}
5	4	pweqi 4552	. . 3 ⊢ 𝒫 3o = 𝒫 {∅, 1o, 2o}
6	3, 5	eleqtrri 2839	. 2 ⊢ {∅} ∈ 𝒫 3o
7	1	a1i 11	. . . . 5 ⊢ (⊤ → {∅, 1o, 2o} ∈ V)
8	2	a1i 11	. . . . . . 7 ⊢ (⊤ → {∅} ⊆ {∅, 1o, 2o})
9		0ex 5232	. . . . . . . 8 ⊢ ∅ ∈ V
10	9	snss 4720	. . . . . . 7 ⊢ (∅ ∈ {∅, 1o, 2o} ↔ {∅} ⊆ {∅, 1o, 2o})
11	8, 10	sylibr 233	. . . . . 6 ⊢ (⊤ → ∅ ∈ {∅, 1o, 2o})
12		snsstp3 4752	. . . . . . . 8 ⊢ {2o} ⊆ {∅, 1o, 2o}
13	12	a1i 11	. . . . . . 7 ⊢ (⊤ → {2o} ⊆ {∅, 1o, 2o})
14		2oex 8317	. . . . . . . 8 ⊢ 2o ∈ V
15	14	snss 4720	. . . . . . 7 ⊢ (2o ∈ {∅, 1o, 2o} ↔ {2o} ⊆ {∅, 1o, 2o})
16	13, 15	sylibr 233	. . . . . 6 ⊢ (⊤ → 2o ∈ {∅, 1o, 2o})
17	11, 16	prssd 4756	. . . . 5 ⊢ (⊤ → {∅, 2o} ⊆ {∅, 1o, 2o})
18	7, 17	sselpwd 5251	. . . 4 ⊢ (⊤ → {∅, 2o} ∈ 𝒫 {∅, 1o, 2o})
19	18	mptru 1546	. . 3 ⊢ {∅, 2o} ∈ 𝒫 {∅, 1o, 2o}
20	19, 5	eleqtrri 2839	. 2 ⊢ {∅, 2o} ∈ 𝒫 3o
21		simpl 483	. . 3 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → {∅} ∈ 𝒫 3o)
22		sseq1 3947	. . . . . 6 ⊢ (𝑠 = {∅} → (𝑠 ⊆ 𝑡 ↔ {∅} ⊆ 𝑡))
23		fveq2 6783	. . . . . . . 8 ⊢ (𝑠 = {∅} → (𝐾‘𝑠) = (𝐾‘{∅}))
24	23	sseq1d 3953	. . . . . . 7 ⊢ (𝑠 = {∅} → ((𝐾‘𝑠) ⊆ (𝐾‘𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)))
25	24	notbid 318	. . . . . 6 ⊢ (𝑠 = {∅} → (¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)))
26	22, 25	anbi12d 631	. . . . 5 ⊢ (𝑠 = {∅} → ((𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) ↔ ({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡))))
27	26	rexbidv 3227	. . . 4 ⊢ (𝑠 = {∅} → (∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) ↔ ∃𝑡 ∈ 𝒫 3o({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡))))
28	27	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))))
29		simpr 485	. . . 4 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → {∅, 2o} ∈ 𝒫 3o)
30		fveq2 6783	. . . . . . . 8 ⊢ (𝑡 = {∅, 2o} → (𝐾‘𝑡) = (𝐾‘{∅, 2o}))
31	30	sseq2d 3954	. . . . . . 7 ⊢ (𝑡 = {∅, 2o} → ((𝐾‘{∅}) ⊆ (𝐾‘𝑡) ↔ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o})))
32	31	notbid 318	. . . . . 6 ⊢ (𝑡 = {∅, 2o} → (¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡) ↔ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o})))
33	32	cleq2lem 41223	. . . . 5 ⊢ (𝑡 = {∅, 2o} → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)) ↔ ({∅} ⊆ {∅, 2o} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))))
34	33	adantl 482	. . . 4 ⊢ ((({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) ∧ 𝑡 = {∅, 2o}) → (({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)) ↔ ({∅} ⊆ {∅, 2o} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))))
35		1oex 8316	. . . . . . . 8 ⊢ 1o ∈ V
36	35	prid2 4700	. . . . . . 7 ⊢ 1o ∈ {∅, 1o}
37		iftrue 4466	. . . . . . . . 9 ⊢ (𝑟 = {∅} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = {∅, 1o})
38		clsk1indlem.k	. . . . . . . . 9 ⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
39		prex 5356	. . . . . . . . 9 ⊢ {∅, 1o} ∈ V
40	37, 38, 39	fvmpt 6884	. . . . . . . 8 ⊢ ({∅} ∈ 𝒫 3o → (𝐾‘{∅}) = {∅, 1o})
41	40	adantr 481	. . . . . . 7 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → (𝐾‘{∅}) = {∅, 1o})
42	36, 41	eleqtrrid 2847	. . . . . 6 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → 1o ∈ (𝐾‘{∅}))
43		1n0 8327	. . . . . . . . . . 11 ⊢ 1o ≠ ∅
44	43	neii 2946	. . . . . . . . . 10 ⊢ ¬ 1o = ∅
45		eqcom 2746	. . . . . . . . . . . 12 ⊢ (1o = 2o ↔ 2o = 1o)
46		df-2o 8307	. . . . . . . . . . . . 13 ⊢ 2o = suc 1o
47		df-1o 8306	. . . . . . . . . . . . 13 ⊢ 1o = suc ∅
48	46, 47	eqeq12i 2757	. . . . . . . . . . . 12 ⊢ (2o = 1o ↔ suc 1o = suc ∅)
49		suc11reg 9386	. . . . . . . . . . . 12 ⊢ (suc 1o = suc ∅ ↔ 1o = ∅)
50	45, 48, 49	3bitri 297	. . . . . . . . . . 11 ⊢ (1o = 2o ↔ 1o = ∅)
51	43, 50	nemtbir 3041	. . . . . . . . . 10 ⊢ ¬ 1o = 2o
52	44, 51	pm3.2ni 878	. . . . . . . . 9 ⊢ ¬ (1o = ∅ ∨ 1o = 2o)
53		elpri 4584	. . . . . . . . 9 ⊢ (1o ∈ {∅, 2o} → (1o = ∅ ∨ 1o = 2o))
54	52, 53	mto 196	. . . . . . . 8 ⊢ ¬ 1o ∈ {∅, 2o}
55	54	a1i 11	. . . . . . 7 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ 1o ∈ {∅, 2o})
56		eqeq1 2743	. . . . . . . . . . 11 ⊢ (𝑟 = {∅, 2o} → (𝑟 = {∅} ↔ {∅, 2o} = {∅}))
57		id 22	. . . . . . . . . . 11 ⊢ (𝑟 = {∅, 2o} → 𝑟 = {∅, 2o})
58	56, 57	ifbieq2d 4486	. . . . . . . . . 10 ⊢ (𝑟 = {∅, 2o} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if({∅, 2o} = {∅}, {∅, 1o}, {∅, 2o}))
59	14	prid2 4700	. . . . . . . . . . . 12 ⊢ 2o ∈ {∅, 2o}
60		2on0 8322	. . . . . . . . . . . . 13 ⊢ 2o ≠ ∅
61		nelsn 4602	. . . . . . . . . . . . 13 ⊢ (2o ≠ ∅ → ¬ 2o ∈ {∅})
62	60, 61	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 2o ∈ {∅}
63		nelneq2 2865	. . . . . . . . . . . 12 ⊢ ((2o ∈ {∅, 2o} ∧ ¬ 2o ∈ {∅}) → ¬ {∅, 2o} = {∅})
64	59, 62, 63	mp2an 689	. . . . . . . . . . 11 ⊢ ¬ {∅, 2o} = {∅}
65	64	iffalsei 4470	. . . . . . . . . 10 ⊢ if({∅, 2o} = {∅}, {∅, 1o}, {∅, 2o}) = {∅, 2o}
66	58, 65	eqtrdi 2795	. . . . . . . . 9 ⊢ (𝑟 = {∅, 2o} → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = {∅, 2o})
67		prex 5356	. . . . . . . . 9 ⊢ {∅, 2o} ∈ V
68	66, 38, 67	fvmpt 6884	. . . . . . . 8 ⊢ ({∅, 2o} ∈ 𝒫 3o → (𝐾‘{∅, 2o}) = {∅, 2o})
69	68	adantl 482	. . . . . . 7 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → (𝐾‘{∅, 2o}) = {∅, 2o})
70	55, 69	neleqtrrd 2862	. . . . . 6 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ 1o ∈ (𝐾‘{∅, 2o}))
71		nelss 3985	. . . . . 6 ⊢ ((1o ∈ (𝐾‘{∅}) ∧ ¬ 1o ∈ (𝐾‘{∅, 2o})) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))
72	42, 70, 71	syl2anc 584	. . . . 5 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o}))
73		snsspr1 4748	. . . . 5 ⊢ {∅} ⊆ {∅, 2o}
74	72, 73	jctil 520	. . . 4 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ({∅} ⊆ {∅, 2o} ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘{∅, 2o})))
75	29, 34, 74	rspcedvd 3564	. . 3 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ∃𝑡 ∈ 𝒫 3o({∅} ⊆ 𝑡 ∧ ¬ (𝐾‘{∅}) ⊆ (𝐾‘𝑡)))
76	21, 28, 75	rspcedvd 3564	. 2 ⊢ (({∅} ∈ 𝒫 3o ∧ {∅, 2o} ∈ 𝒫 3o) → ∃𝑠 ∈ 𝒫 3o∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)))
77	6, 20, 76	mp2an 689	1 ⊢ ∃𝑠 ∈ 𝒫 3o∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539  ⊤wtru 1540   ∈ wcel 2107   ≠ wne 2944  ∃wrex 3066  Vcvv 3433   ⊆ wss 3888  ∅c0 4257  ifcif 4460  𝒫 cpw 4534  {csn 4562  {cpr 4564  {ctp 4566   ↦ cmpt 5158  suc csuc 6272  ‘cfv 6437  1_oc1o 8299  2_oc2o 8300  3_oc3o 8301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-sep 5224  ax-nul 5231  ax-pr 5353  ax-un 7597  ax-reg 9360

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3435  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4841  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5490  df-eprel 5496  df-fr 5545  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-suc 6276  df-iota 6395  df-fun 6439  df-fv 6445  df-1o 8306  df-2o 8307  df-3o 8308

This theorem is referenced by:  clsk1independent  41663