Theorem clsk1independent 44627

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 8454	. . 3 ⊢ 3o ∈ On
2	1	elexi 3477	. 2 ⊢ 3o ∈ V
3		eqid 2763	. . . . 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 6428	. . . . . . . . . . . . 13 ⊢ 2o ⊆ suc 2o
7		2oex 8449	. . . . . . . . . . . . . 14 ⊢ 2o ∈ V
8	7	elpw 4560	. . . . . . . . . . . . 13 ⊢ (2o ∈ 𝒫 suc 2o ↔ 2o ⊆ suc 2o)
9	6, 8	mpbir 233	. . . . . . . . . . . 12 ⊢ 2o ∈ 𝒫 suc 2o
10		df2o3 8445	. . . . . . . . . . . 12 ⊢ 2o = {∅, 1o}
11		df-3o 8439	. . . . . . . . . . . . . 14 ⊢ 3o = suc 2o
12	11	eqcomi 2772	. . . . . . . . . . . . 13 ⊢ suc 2o = 3o
13	12	pweqi 4572	. . . . . . . . . . . 12 ⊢ 𝒫 suc 2o = 𝒫 3o
14	9, 10, 13	3eltr3i 2875	. . . . . . . . . . 11 ⊢ {∅, 1o} ∈ 𝒫 3o
15	14	2a1i 12	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝒫 3o → (¬ ¬ 𝑟 = {∅} → {∅, 1o} ∈ 𝒫 3o))
16	5, 15	jcad 520	. . . . . . . . 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 564	. . . . . . 7 ⊢ (𝑟 ∈ 𝒫 3o → (¬ (𝑟 = {∅} ∧ {∅, 1o} ∈ 𝒫 3o) → (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3o)))
19	18	orrd 874	. . . . . 6 ⊢ (𝑟 ∈ 𝒫 3o → ((𝑟 = {∅} ∧ {∅, 1o} ∈ 𝒫 3o) ∨ (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3o)))
20		ifel 4526	. . . . . 6 ⊢ (if(𝑟 = {∅}, {∅, 1o}, 𝑟) ∈ 𝒫 3o ↔ ((𝑟 = {∅} ∧ {∅, 1o} ∈ 𝒫 3o) ∨ (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3o)))
21	19, 20	sylibr 236	. . . . 5 ⊢ (𝑟 ∈ 𝒫 3o → if(𝑟 = {∅}, {∅, 1o}, 𝑟) ∈ 𝒫 3o)
22	3, 21	fmpti 7093	. . . 4 ⊢ (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)):𝒫 3o⟶𝒫 3o
23	2	pwex 5338	. . . . 5 ⊢ 𝒫 3o ∈ V
24	23, 23	elmap 8853	. . . 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 233	. . 3 ⊢ (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) ∈ (𝒫 3o ↑m 𝒫 3o)
26	3	clsk1indlem0 44622	. . . . . 6 ⊢ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅
27	3	clsk1indlem2 44623	. . . . . 6 ⊢ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)
28	26, 27	pm3.2i 474	. . . . 5 ⊢ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠))
29	3	clsk1indlem3 44624	. . . . . 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 44625	. . . . . 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 474	. . . . 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 474	. . . 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 44626	. . . 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 474	. . 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 6866	. . . . . . . 8 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘∅) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅))
36	35	eqeq1d 2765	. . . . . . 7 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → ((𝑘‘∅) = ∅ ↔ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅))
37		fveq1 6866	. . . . . . . . 9 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘𝑠) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠))
38	37	sseq2d 3969	. . . . . . . 8 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑠 ⊆ (𝑘‘𝑠) ↔ 𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)))
39	38	ralbidv 3186	. . . . . . 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 641	. . . . . 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 6866	. . . . . . . . 9 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘(𝑠 ∪ 𝑡)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘(𝑠 ∪ 𝑡)))
42		fveq1 6866	. . . . . . . . . 10 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘𝑡) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡))
43	37, 42	uneq12d 4123	. . . . . . . . 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 3970	. . . . . . . 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 3227	. . . . . . 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 6874	. . . . . . . . 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 2779	. . . . . . . 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 3186	. . . . . . 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 641	. . . . . 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 641	. . . . 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 3145	. . . . . 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		pm4.61 408	. . . . . . . 8 ⊢ (¬ (𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)) ↔ (𝑠 ⊆ 𝑡 ∧ ¬ (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))
54	37, 42	sseq12d 3970	. . . . . . . . . 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 320	. . . . . . . . 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 639	. . . . . . . 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 285	. . . . . . 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 3228	. . . . . 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 287	. . . . 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 641	. . . 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 3582	. . 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 702	. 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 4570	. . . . . 6 ⊢ (𝑏 = 3o → 𝒫 𝑏 = 𝒫 3o)
64	63, 63	oveq12d 7414	. . . . 5 ⊢ (𝑏 = 3o → (𝒫 𝑏 ↑m 𝒫 𝑏) = (𝒫 3o ↑m 𝒫 3o))
65		pm4.61 408	. . . . . 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 3321	. . . . . . . . . 10 ⊢ (𝑏 = 3o → (∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘‘𝑠) ↔ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)))
70	68, 69	bitrid 285	. . . . . . . . 9 ⊢ (𝑏 = 3o → (𝜒 ↔ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)))
71	67, 70	anbi12d 641	. . . . . . . 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 3321	. . . . . . . . . . 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 3337	. . . . . . . . . 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 285	. . . . . . . . 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 3321	. . . . . . . . . 10 ⊢ (𝑏 = 3o → (∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠) ↔ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠)))
78	76, 77	bitrid 285	. . . . . . . . 9 ⊢ (𝑏 = 3o → (𝜏 ↔ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠)))
79	75, 78	anbi12d 641	. . . . . . . 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 641	. . . . . . 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 3321	. . . . . . . . . 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 3337	. . . . . . . . 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 285	. . . . . . . 8 ⊢ (𝑏 = 3o → (𝜓 ↔ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))))
85	84	notbid 320	. . . . . . 7 ⊢ (𝑏 = 3o → (¬ 𝜓 ↔ ¬ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))))
86	80, 85	anbi12d 641	. . . . . 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 285	. . . . 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 3338	. . . 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 3582	. . 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 3145	. . . 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 3481	. . . 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 336	. . 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 220	. 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 702	1 ⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858  ∀wal 1559   = wceq 1561   ∈ wcel 2143  ∀wral 3077  ∃wrex 3087  Vcvv 3455   ∪ cun 3903   ⊆ wss 3905  ∅c0 4286  ifcif 4481  𝒫 cpw 4556  {csn 4583  {cpr 4585   ↦ cmpt 5182  Oncon0 6346  suc csuc 6348  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  1_oc1o 8430  2_oc2o 8431  3_oc3o 8432   ↑_m cmap 8808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718  ax-reg 9538

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-xor 1533  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-ord 6349  df-on 6350  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1o 8437  df-2o 8438  df-3o 8439  df-map 8810

This theorem is referenced by: (None)