Theorem clsk1independent 44144

Description: For generalized closure functions, property K1 (isotony) is independent of the properties K0, K2, K3, K4. This contradicts a claim which appears in preprints of Table 2 in Bärbel M. R. Stadler and Peter F. Stadler. "Generalized Topological Spaces in Evolutionary Theory and Combinatorial Chemistry." J. Chem. Inf. Comput. Sci., 42:577-585, 2002. Proceedings MCC 2001, Dubrovnik. The same table row implying K1 follows from the other four appears in the supplemental materials Bärbel M. R. Stadler and Peter F. Stadler. "Basic Properties of Closure Spaces" 2001 on page 12. (Contributed by RP, 5-Jul-2021.)

Hypotheses

Ref	Expression
clsnim.k0	⊢ (𝜑 ↔ (𝑘‘∅) = ∅)
clsnim.k1	⊢ (𝜓 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))
clsnim.k2	⊢ (𝜒 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘‘𝑠))
clsnim.k3	⊢ (𝜃 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)))
clsnim.k4	⊢ (𝜏 ↔ ∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))

Assertion

Ref	Expression
clsk1independent	⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓)

Distinct variable group:   𝑘,𝑏,𝑠,𝑡

Allowed substitution hints:   𝜑(𝑡,𝑘,𝑠,𝑏)   𝜓(𝑡,𝑘,𝑠,𝑏)   𝜒(𝑡,𝑘,𝑠,𝑏)   𝜃(𝑡,𝑘,𝑠,𝑏)   𝜏(𝑡,𝑘,𝑠,𝑏)

Proof of Theorem clsk1independent

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3on 8407	. . 3 ⊢ 3o ∈ On
2	1	elexi 3459	. 2 ⊢ 3o ∈ V
3		eqid 2731	. . . . 5 ⊢ (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
4		notnotr 130	. . . . . . . . . . 11 ⊢ (¬ ¬ 𝑟 = {∅} → 𝑟 = {∅})
5	4	a1i 11	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝒫 3o → (¬ ¬ 𝑟 = {∅} → 𝑟 = {∅}))
6		sssucid 6394	. . . . . . . . . . . . 13 ⊢ 2o ⊆ suc 2o
7		2oex 8402	. . . . . . . . . . . . . 14 ⊢ 2o ∈ V
8	7	elpw 4553	. . . . . . . . . . . . 13 ⊢ (2o ∈ 𝒫 suc 2o ↔ 2o ⊆ suc 2o)
9	6, 8	mpbir 231	. . . . . . . . . . . 12 ⊢ 2o ∈ 𝒫 suc 2o
10		df2o3 8399	. . . . . . . . . . . 12 ⊢ 2o = {∅, 1o}
11		df-3o 8393	. . . . . . . . . . . . . 14 ⊢ 3o = suc 2o
12	11	eqcomi 2740	. . . . . . . . . . . . 13 ⊢ suc 2o = 3o
13	12	pweqi 4565	. . . . . . . . . . . 12 ⊢ 𝒫 suc 2o = 𝒫 3o
14	9, 10, 13	3eltr3i 2843	. . . . . . . . . . 11 ⊢ {∅, 1o} ∈ 𝒫 3o
15	14	2a1i 12	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝒫 3o → (¬ ¬ 𝑟 = {∅} → {∅, 1o} ∈ 𝒫 3o))
16	5, 15	jcad 512	. . . . . . . . 9 ⊢ (𝑟 ∈ 𝒫 3o → (¬ ¬ 𝑟 = {∅} → (𝑟 = {∅} ∧ {∅, 1o} ∈ 𝒫 3o)))
17	16	con1d 145	. . . . . . . 8 ⊢ (𝑟 ∈ 𝒫 3o → (¬ (𝑟 = {∅} ∧ {∅, 1o} ∈ 𝒫 3o) → ¬ 𝑟 = {∅}))
18	17	anc2ri 556	. . . . . . 7 ⊢ (𝑟 ∈ 𝒫 3o → (¬ (𝑟 = {∅} ∧ {∅, 1o} ∈ 𝒫 3o) → (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3o)))
19	18	orrd 863	. . . . . 6 ⊢ (𝑟 ∈ 𝒫 3o → ((𝑟 = {∅} ∧ {∅, 1o} ∈ 𝒫 3o) ∨ (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3o)))
20		ifel 4519	. . . . . 6 ⊢ (if(𝑟 = {∅}, {∅, 1o}, 𝑟) ∈ 𝒫 3o ↔ ((𝑟 = {∅} ∧ {∅, 1o} ∈ 𝒫 3o) ∨ (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3o)))
21	19, 20	sylibr 234	. . . . 5 ⊢ (𝑟 ∈ 𝒫 3o → if(𝑟 = {∅}, {∅, 1o}, 𝑟) ∈ 𝒫 3o)
22	3, 21	fmpti 7051	. . . 4 ⊢ (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)):𝒫 3o⟶𝒫 3o
23	2	pwex 5320	. . . . 5 ⊢ 𝒫 3o ∈ V
24	23, 23	elmap 8801	. . . 4 ⊢ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) ∈ (𝒫 3o ↑m 𝒫 3o) ↔ (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)):𝒫 3o⟶𝒫 3o)
25	22, 24	mpbir 231	. . 3 ⊢ (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) ∈ (𝒫 3o ↑m 𝒫 3o)
26	3	clsk1indlem0 44139	. . . . . 6 ⊢ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅
27	3	clsk1indlem2 44140	. . . . . 6 ⊢ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)
28	26, 27	pm3.2i 470	. . . . 5 ⊢ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠))
29	3	clsk1indlem3 44141	. . . . . 6 ⊢ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘(𝑠 ∪ 𝑡)) ⊆ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ∪ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡))
30	3	clsk1indlem4 44142	. . . . . 6 ⊢ ∀𝑠 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)
31	29, 30	pm3.2i 470	. . . . 5 ⊢ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘(𝑠 ∪ 𝑡)) ⊆ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ∪ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠))
32	28, 31	pm3.2i 470	. . . 4 ⊢ ((((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘(𝑠 ∪ 𝑡)) ⊆ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ∪ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)))
33	3	clsk1indlem1 44143	. . . 4 ⊢ ∃𝑠 ∈ 𝒫 3o∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡))
34	32, 33	pm3.2i 470	. . 3 ⊢ (((((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘(𝑠 ∪ 𝑡)) ⊆ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ∪ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠))) ∧ ∃𝑠 ∈ 𝒫 3o∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡)))
35		fveq1 6827	. . . . . . . 8 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘∅) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅))
36	35	eqeq1d 2733	. . . . . . 7 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → ((𝑘‘∅) = ∅ ↔ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅))
37		fveq1 6827	. . . . . . . . 9 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘𝑠) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠))
38	37	sseq2d 3962	. . . . . . . 8 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑠 ⊆ (𝑘‘𝑠) ↔ 𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)))
39	38	ralbidv 3155	. . . . . . 7 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠) ↔ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)))
40	36, 39	anbi12d 632	. . . . . 6 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)) ↔ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠))))
41		fveq1 6827	. . . . . . . . 9 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘(𝑠 ∪ 𝑡)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘(𝑠 ∪ 𝑡)))
42		fveq1 6827	. . . . . . . . . 10 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘𝑡) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡))
43	37, 42	uneq12d 4118	. . . . . . . . 9 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ∪ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡)))
44	41, 43	sseq12d 3963	. . . . . . . 8 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → ((𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ↔ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘(𝑠 ∪ 𝑡)) ⊆ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ∪ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡))))
45	44	2ralbidv 3196	. . . . . . 7 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘(𝑠 ∪ 𝑡)) ⊆ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ∪ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡))))
46		id 22	. . . . . . . . . 10 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → 𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)))
47	46, 37	fveq12d 6835	. . . . . . . . 9 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘(𝑘‘𝑠)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)))
48	47, 37	eqeq12d 2747	. . . . . . . 8 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → ((𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠) ↔ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)))
49	48	ralbidv 3155	. . . . . . 7 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠) ↔ ∀𝑠 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)))
50	45, 49	anbi12d 632	. . . . . 6 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → ((∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠)) ↔ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘(𝑠 ∪ 𝑡)) ⊆ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ∪ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠))))
51	40, 50	anbi12d 632	. . . . 5 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → ((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))) ↔ ((((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘(𝑠 ∪ 𝑡)) ⊆ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ∪ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)))))
52		rexnal2 3114	. . . . . 6 ⊢ (∃𝑠 ∈ 𝒫 3o∃𝑡 ∈ 𝒫 3o ¬ (𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)) ↔ ¬ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))
53		pm4.61 404	. . . . . . . 8 ⊢ (¬ (𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)) ↔ (𝑠 ⊆ 𝑡 ∧ ¬ (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))
54	37, 42	sseq12d 3963	. . . . . . . . . 10 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → ((𝑘‘𝑠) ⊆ (𝑘‘𝑡) ↔ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡)))
55	54	notbid 318	. . . . . . . . 9 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (¬ (𝑘‘𝑠) ⊆ (𝑘‘𝑡) ↔ ¬ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡)))
56	55	anbi2d 630	. . . . . . . 8 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → ((𝑠 ⊆ 𝑡 ∧ ¬ (𝑘‘𝑠) ⊆ (𝑘‘𝑡)) ↔ (𝑠 ⊆ 𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡))))
57	53, 56	bitrid 283	. . . . . . 7 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (¬ (𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)) ↔ (𝑠 ⊆ 𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡))))
58	57	2rexbidv 3197	. . . . . 6 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (∃𝑠 ∈ 𝒫 3o∃𝑡 ∈ 𝒫 3o ¬ (𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)) ↔ ∃𝑠 ∈ 𝒫 3o∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡))))
59	52, 58	bitr3id 285	. . . . 5 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (¬ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)) ↔ ∃𝑠 ∈ 𝒫 3o∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡))))
60	51, 59	anbi12d 632	. . . 4 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))) ↔ (((((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘(𝑠 ∪ 𝑡)) ⊆ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ∪ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠))) ∧ ∃𝑠 ∈ 𝒫 3o∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡)))))
61	60	rspcev 3572	. . 3 ⊢ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) ∈ (𝒫 3o ↑m 𝒫 3o) ∧ (((((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘(𝑠 ∪ 𝑡)) ⊆ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ∪ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠))) ∧ ∃𝑠 ∈ 𝒫 3o∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡)))) → ∃𝑘 ∈ (𝒫 3o ↑m 𝒫 3o)((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))))
62	25, 34, 61	mp2an 692	. 2 ⊢ ∃𝑘 ∈ (𝒫 3o ↑m 𝒫 3o)((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))
63		pweq 4563	. . . . . 6 ⊢ (𝑏 = 3o → 𝒫 𝑏 = 𝒫 3o)
64	63, 63	oveq12d 7370	. . . . 5 ⊢ (𝑏 = 3o → (𝒫 𝑏 ↑m 𝒫 𝑏) = (𝒫 3o ↑m 𝒫 3o))
65		pm4.61 404	. . . . . 6 ⊢ (¬ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓) ↔ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) ∧ ¬ 𝜓))
66		clsnim.k0	. . . . . . . . . 10 ⊢ (𝜑 ↔ (𝑘‘∅) = ∅)
67	66	a1i 11	. . . . . . . . 9 ⊢ (𝑏 = 3o → (𝜑 ↔ (𝑘‘∅) = ∅))
68		clsnim.k2	. . . . . . . . . 10 ⊢ (𝜒 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘‘𝑠))
69	63	raleqdv 3292	. . . . . . . . . 10 ⊢ (𝑏 = 3o → (∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘‘𝑠) ↔ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)))
70	68, 69	bitrid 283	. . . . . . . . 9 ⊢ (𝑏 = 3o → (𝜒 ↔ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)))
71	67, 70	anbi12d 632	. . . . . . . 8 ⊢ (𝑏 = 3o → ((𝜑 ∧ 𝜒) ↔ ((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠))))
72		clsnim.k3	. . . . . . . . . 10 ⊢ (𝜃 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)))
73	63	raleqdv 3292	. . . . . . . . . . 11 ⊢ (𝑏 = 3o → (∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ↔ ∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡))))
74	63, 73	raleqbidv 3312	. . . . . . . . . 10 ⊢ (𝑏 = 3o → (∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡))))
75	72, 74	bitrid 283	. . . . . . . . 9 ⊢ (𝑏 = 3o → (𝜃 ↔ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡))))
76		clsnim.k4	. . . . . . . . . 10 ⊢ (𝜏 ↔ ∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))
77	63	raleqdv 3292	. . . . . . . . . 10 ⊢ (𝑏 = 3o → (∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠) ↔ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠)))
78	76, 77	bitrid 283	. . . . . . . . 9 ⊢ (𝑏 = 3o → (𝜏 ↔ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠)))
79	75, 78	anbi12d 632	. . . . . . . 8 ⊢ (𝑏 = 3o → ((𝜃 ∧ 𝜏) ↔ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))))
80	71, 79	anbi12d 632	. . . . . . 7 ⊢ (𝑏 = 3o → (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) ↔ (((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠)))))
81		clsnim.k1	. . . . . . . . 9 ⊢ (𝜓 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))
82	63	raleqdv 3292	. . . . . . . . . 10 ⊢ (𝑏 = 3o → (∀𝑡 ∈ 𝒫 𝑏(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)) ↔ ∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))))
83	63, 82	raleqbidv 3312	. . . . . . . . 9 ⊢ (𝑏 = 3o → (∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))))
84	81, 83	bitrid 283	. . . . . . . 8 ⊢ (𝑏 = 3o → (𝜓 ↔ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))))
85	84	notbid 318	. . . . . . 7 ⊢ (𝑏 = 3o → (¬ 𝜓 ↔ ¬ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))))
86	80, 85	anbi12d 632	. . . . . 6 ⊢ (𝑏 = 3o → ((((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) ∧ ¬ 𝜓) ↔ ((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))))
87	65, 86	bitrid 283	. . . . 5 ⊢ (𝑏 = 3o → (¬ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓) ↔ ((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))))
88	64, 87	rexeqbidv 3313	. . . 4 ⊢ (𝑏 = 3o → (∃𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ¬ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓) ↔ ∃𝑘 ∈ (𝒫 3o ↑m 𝒫 3o)((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))))
89	88	rspcev 3572	. . 3 ⊢ ((3o ∈ V ∧ ∃𝑘 ∈ (𝒫 3o ↑m 𝒫 3o)((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))) → ∃𝑏 ∈ V ∃𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ¬ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓))
90		rexnal2 3114	. . . 4 ⊢ (∃𝑏 ∈ V ∃𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ¬ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓) ↔ ¬ ∀𝑏 ∈ V ∀𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓))
91		ralv 3463	. . . 4 ⊢ (∀𝑏 ∈ V ∀𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓) ↔ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓))
92	90, 91	xchbinx 334	. . 3 ⊢ (∃𝑏 ∈ V ∃𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ¬ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓) ↔ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓))
93	89, 92	sylib 218	. 2 ⊢ ((3o ∈ V ∧ ∃𝑘 ∈ (𝒫 3o ↑m 𝒫 3o)((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))) → ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓))
94	2, 62, 93	mp2an 692	1 ⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847  ∀wal 1539   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∪ cun 3895   ⊆ wss 3897  ∅c0 4282  ifcif 4474  𝒫 cpw 4549  {csn 4575  {cpr 4577   ↦ cmpt 5174  Oncon0 6312  suc csuc 6314  ⟶wf 6483  ‘cfv 6487  (class class class)co 7352  1_oc1o 8384  2_oc2o 8385  3_oc3o 8386   ↑_m cmap 8756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-reg 9484

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1513  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6315  df-on 6316  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fv 6495  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1o 8391  df-2o 8392  df-3o 8393  df-map 8758

This theorem is referenced by: (None)