Theorem clsk1independent 44323

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 8415	. . 3 ⊢ 3o ∈ On
2	1	elexi 3464	. 2 ⊢ 3o ∈ V
3		eqid 2737	. . . . 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 6400	. . . . . . . . . . . . 13 ⊢ 2o ⊆ suc 2o
7		2oex 8410	. . . . . . . . . . . . . 14 ⊢ 2o ∈ V
8	7	elpw 4559	. . . . . . . . . . . . 13 ⊢ (2o ∈ 𝒫 suc 2o ↔ 2o ⊆ suc 2o)
9	6, 8	mpbir 231	. . . . . . . . . . . 12 ⊢ 2o ∈ 𝒫 suc 2o
10		df2o3 8407	. . . . . . . . . . . 12 ⊢ 2o = {∅, 1o}
11		df-3o 8401	. . . . . . . . . . . . . 14 ⊢ 3o = suc 2o
12	11	eqcomi 2746	. . . . . . . . . . . . 13 ⊢ suc 2o = 3o
13	12	pweqi 4571	. . . . . . . . . . . 12 ⊢ 𝒫 suc 2o = 𝒫 3o
14	9, 10, 13	3eltr3i 2849	. . . . . . . . . . 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 864	. . . . . 6 ⊢ (𝑟 ∈ 𝒫 3o → ((𝑟 = {∅} ∧ {∅, 1o} ∈ 𝒫 3o) ∨ (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3o)))
20		ifel 4525	. . . . . 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 7059	. . . 4 ⊢ (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)):𝒫 3o⟶𝒫 3o
23	2	pwex 5326	. . . . 5 ⊢ 𝒫 3o ∈ V
24	23, 23	elmap 8813	. . . 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 44318	. . . . . 6 ⊢ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅
27	3	clsk1indlem2 44319	. . . . . 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 44320	. . . . . 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 44321	. . . . . 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 44322	. . . 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 6834	. . . . . . . 8 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘∅) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅))
36	35	eqeq1d 2739	. . . . . . 7 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → ((𝑘‘∅) = ∅ ↔ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅))
37		fveq1 6834	. . . . . . . . 9 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘𝑠) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠))
38	37	sseq2d 3967	. . . . . . . 8 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑠 ⊆ (𝑘‘𝑠) ↔ 𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)))
39	38	ralbidv 3160	. . . . . . 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 633	. . . . . 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 6834	. . . . . . . . 9 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘(𝑠 ∪ 𝑡)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘(𝑠 ∪ 𝑡)))
42		fveq1 6834	. . . . . . . . . 10 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘𝑡) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡))
43	37, 42	uneq12d 4122	. . . . . . . . 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 3968	. . . . . . . 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 3201	. . . . . . 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 6842	. . . . . . . . 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 2753	. . . . . . . 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 3160	. . . . . . 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 633	. . . . . 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 633	. . . . 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 3119	. . . . . 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 3968	. . . . . . . . . 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 631	. . . . . . . 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 3202	. . . . . 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 633	. . . 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 3577	. . 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 693	. 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 4569	. . . . . 6 ⊢ (𝑏 = 3o → 𝒫 𝑏 = 𝒫 3o)
64	63, 63	oveq12d 7378	. . . . 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 3297	. . . . . . . . . 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 633	. . . . . . . 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 3297	. . . . . . . . . . 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 3317	. . . . . . . . . 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 3297	. . . . . . . . . 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 633	. . . . . . . 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 633	. . . . . . 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 3297	. . . . . . . . . 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 3317	. . . . . . . . 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 633	. . . . . 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 3318	. . . 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 3577	. . 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 3119	. . . 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 3468	. . . 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 693	1 ⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061  Vcvv 3441   ∪ cun 3900   ⊆ wss 3902  ∅c0 4286  ifcif 4480  𝒫 cpw 4555  {csn 4581  {cpr 4583   ↦ cmpt 5180  Oncon0 6318  suc csuc 6320  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360  1_oc1o 8392  2_oc2o 8393  3_oc3o 8394   ↑_m cmap 8767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-reg 9501

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1514  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1o 8399  df-2o 8400  df-3o 8401  df-map 8769

This theorem is referenced by: (None)