Theorem clsk1independent 38344

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 7570	. . 3 ⊢ 3𝑜 ∈ On
2	1	elexi 3213	. 2 ⊢ 3𝑜 ∈ V
3		eqid 2622	. . . . 5 ⊢ (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
4		notnotr 125	. . . . . . . . . . 11 ⊢ (¬ ¬ 𝑟 = {∅} → 𝑟 = {∅})
5	4	a1i 11	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝒫 3𝑜 → (¬ ¬ 𝑟 = {∅} → 𝑟 = {∅}))
6		sssucid 5802	. . . . . . . . . . . . 13 ⊢ 2𝑜 ⊆ suc 2𝑜
7		2on 7568	. . . . . . . . . . . . . . 15 ⊢ 2𝑜 ∈ On
8	7	elexi 3213	. . . . . . . . . . . . . 14 ⊢ 2𝑜 ∈ V
9	8	elpw 4164	. . . . . . . . . . . . 13 ⊢ (2𝑜 ∈ 𝒫 suc 2𝑜 ↔ 2𝑜 ⊆ suc 2𝑜)
10	6, 9	mpbir 221	. . . . . . . . . . . 12 ⊢ 2𝑜 ∈ 𝒫 suc 2𝑜
11		df2o3 7573	. . . . . . . . . . . 12 ⊢ 2𝑜 = {∅, 1𝑜}
12		df-3o 7562	. . . . . . . . . . . . . 14 ⊢ 3𝑜 = suc 2𝑜
13	12	eqcomi 2631	. . . . . . . . . . . . 13 ⊢ suc 2𝑜 = 3𝑜
14	13	pweqi 4162	. . . . . . . . . . . 12 ⊢ 𝒫 suc 2𝑜 = 𝒫 3𝑜
15	10, 11, 14	3eltr3i 2713	. . . . . . . . . . 11 ⊢ {∅, 1𝑜} ∈ 𝒫 3𝑜
16	15	2a1i 12	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝒫 3𝑜 → (¬ ¬ 𝑟 = {∅} → {∅, 1𝑜} ∈ 𝒫 3𝑜))
17	5, 16	jcad 555	. . . . . . . . 9 ⊢ (𝑟 ∈ 𝒫 3𝑜 → (¬ ¬ 𝑟 = {∅} → (𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜)))
18	17	con1d 139	. . . . . . . 8 ⊢ (𝑟 ∈ 𝒫 3𝑜 → (¬ (𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜) → ¬ 𝑟 = {∅}))
19	18	anc2ri 581	. . . . . . 7 ⊢ (𝑟 ∈ 𝒫 3𝑜 → (¬ (𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜) → (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3𝑜)))
20	19	orrd 393	. . . . . 6 ⊢ (𝑟 ∈ 𝒫 3𝑜 → ((𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜) ∨ (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3𝑜)))
21		ifel 4129	. . . . . 6 ⊢ (if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) ∈ 𝒫 3𝑜 ↔ ((𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜) ∨ (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3𝑜)))
22	20, 21	sylibr 224	. . . . 5 ⊢ (𝑟 ∈ 𝒫 3𝑜 → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) ∈ 𝒫 3𝑜)
23	3, 22	fmpti 6383	. . . 4 ⊢ (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)):𝒫 3𝑜⟶𝒫 3𝑜
24	2	pwex 4848	. . . . 5 ⊢ 𝒫 3𝑜 ∈ V
25	24, 24	elmap 7886	. . . 4 ⊢ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) ∈ (𝒫 3𝑜 ↑𝑚 𝒫 3𝑜) ↔ (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)):𝒫 3𝑜⟶𝒫 3𝑜)
26	23, 25	mpbir 221	. . 3 ⊢ (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) ∈ (𝒫 3𝑜 ↑𝑚 𝒫 3𝑜)
27	3	clsk1indlem0 38339	. . . . . 6 ⊢ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅
28	3	clsk1indlem2 38340	. . . . . 6 ⊢ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)
29	27, 28	pm3.2i 471	. . . . 5 ⊢ (((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠))
30	3	clsk1indlem3 38341	. . . . . 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))
31	3	clsk1indlem4 38342	. . . . . 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)
32	30, 31	pm3.2i 471	. . . . 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))
33	29, 32	pm3.2i 471	. . . 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)))
34	3	clsk1indlem1 38343	. . . 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))
35	33, 34	pm3.2i 471	. . 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)))
36		fveq1 6190	. . . . . . . 8 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑘‘∅) = ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅))
37	36	eqeq1d 2624	. . . . . . 7 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → ((𝑘‘∅) = ∅ ↔ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅))
38		fveq1 6190	. . . . . . . . 9 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑘‘𝑠) = ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠))
39	38	sseq2d 3633	. . . . . . . 8 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑠 ⊆ (𝑘‘𝑠) ↔ 𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)))
40	39	ralbidv 2986	. . . . . . 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)))
41	37, 40	anbi12d 747	. . . . . 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))))
42		fveq1 6190	. . . . . . . . 9 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑘‘(𝑠 ∪ 𝑡)) = ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘(𝑠 ∪ 𝑡)))
43		fveq1 6190	. . . . . . . . . 10 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑘‘𝑡) = ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡))
44	38, 43	uneq12d 3768	. . . . . . . . 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)))
45	42, 44	sseq12d 3634	. . . . . . . 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))))
46	45	2ralbidv 2989	. . . . . . 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))))
47		id 22	. . . . . . . . . 10 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → 𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)))
48	47, 38	fveq12d 6197	. . . . . . . . 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)))
49	48, 38	eqeq12d 2637	. . . . . . . 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)))
50	49	ralbidv 2986	. . . . . . 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)))
51	46, 50	anbi12d 747	. . . . . 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))))
52	41, 51	anbi12d 747	. . . . 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)))))
53		rexnal2 3043	. . . . . 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)))
54		pm4.61 442	. . . . . . . 8 ⊢ (¬ (𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)) ↔ (𝑠 ⊆ 𝑡 ∧ ¬ (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))
55	38, 43	sseq12d 3634	. . . . . . . . . 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)))
56	55	notbid 308	. . . . . . . . 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)))
57	56	anbi2d 740	. . . . . . . 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))))
58	54, 57	syl5bb 272	. . . . . . 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))))
59	58	2rexbidv 3057	. . . . . 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))))
60	53, 59	syl5bbr 274	. . . . 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))))
61	52, 60	anbi12d 747	. . . 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)))))
62	61	rspcev 3309	. . 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))))
63	26, 35, 62	mp2an 708	. 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)))
64		pweq 4161	. . . . . 6 ⊢ (𝑏 = 3𝑜 → 𝒫 𝑏 = 𝒫 3𝑜)
65	64, 64	oveq12d 6668	. . . . 5 ⊢ (𝑏 = 3𝑜 → (𝒫 𝑏 ↑𝑚 𝒫 𝑏) = (𝒫 3𝑜 ↑𝑚 𝒫 3𝑜))
66		pm4.61 442	. . . . . 6 ⊢ (¬ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓) ↔ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) ∧ ¬ 𝜓))
67		clsnim.k0	. . . . . . . . . 10 ⊢ (𝜑 ↔ (𝑘‘∅) = ∅)
68	67	a1i 11	. . . . . . . . 9 ⊢ (𝑏 = 3𝑜 → (𝜑 ↔ (𝑘‘∅) = ∅))
69		clsnim.k2	. . . . . . . . . 10 ⊢ (𝜒 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘‘𝑠))
70	64	raleqdv 3144	. . . . . . . . . 10 ⊢ (𝑏 = 3𝑜 → (∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘‘𝑠) ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘‘𝑠)))
71	69, 70	syl5bb 272	. . . . . . . . 9 ⊢ (𝑏 = 3𝑜 → (𝜒 ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘‘𝑠)))
72	68, 71	anbi12d 747	. . . . . . . 8 ⊢ (𝑏 = 3𝑜 → ((𝜑 ∧ 𝜒) ↔ ((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘‘𝑠))))
73		clsnim.k3	. . . . . . . . . 10 ⊢ (𝜃 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)))
74	64	raleqdv 3144	. . . . . . . . . . 11 ⊢ (𝑏 = 3𝑜 → (∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) ↔ ∀𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡))))
75	64, 74	raleqbidv 3152	. . . . . . . . . 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))))
76	73, 75	syl5bb 272	. . . . . . . . 9 ⊢ (𝑏 = 3𝑜 → (𝜃 ↔ ∀𝑠 ∈ 𝒫 3𝑜∀𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡))))
77		clsnim.k4	. . . . . . . . . 10 ⊢ (𝜏 ↔ ∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))
78	64	raleqdv 3144	. . . . . . . . . 10 ⊢ (𝑏 = 3𝑜 → (∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠) ↔ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠)))
79	77, 78	syl5bb 272	. . . . . . . . 9 ⊢ (𝑏 = 3𝑜 → (𝜏 ↔ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠)))
80	76, 79	anbi12d 747	. . . . . . . 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))))
81	72, 80	anbi12d 747	. . . . . . 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)))))
82		clsnim.k1	. . . . . . . . 9 ⊢ (𝜓 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))
83	64	raleqdv 3144	. . . . . . . . . 10 ⊢ (𝑏 = 3𝑜 → (∀𝑡 ∈ 𝒫 𝑏(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)) ↔ ∀𝑡 ∈ 𝒫 3𝑜(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))))
84	64, 83	raleqbidv 3152	. . . . . . . . 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))))
85	82, 84	syl5bb 272	. . . . . . . 8 ⊢ (𝑏 = 3𝑜 → (𝜓 ↔ ∀𝑠 ∈ 𝒫 3𝑜∀𝑡 ∈ 𝒫 3𝑜(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))))
86	85	notbid 308	. . . . . . 7 ⊢ (𝑏 = 3𝑜 → (¬ 𝜓 ↔ ¬ ∀𝑠 ∈ 𝒫 3𝑜∀𝑡 ∈ 𝒫 3𝑜(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))))
87	81, 86	anbi12d 747	. . . . . 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)))))
88	66, 87	syl5bb 272	. . . . 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)))))
89	65, 88	rexeqbidv 3153	. . . 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)))))
90	89	rspcev 3309	. . 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))
91		rexnal2 3043	. . . 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))
92		ralv 3219	. . . 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))
93	91, 92	xchbinx 324	. . 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))
94	90, 93	sylib 208	. 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))
95	2, 63, 94	mp2an 708	1 ⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  {csn 4177  {cpr 4179   ↦ cmpt 4729  Oncon0 5723  suc csuc 5725  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553  2_𝑜c2o 7554  3_𝑜c3o 7555   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-xor 1465  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-2o 7561  df-3o 7562  df-map 7859

This theorem is referenced by: (None)