Theorem clsk1independent 39179

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 7842	. . 3 ⊢ 3o ∈ On
2	1	elexi 3430	. 2 ⊢ 3o ∈ V
3		eqid 2825	. . . . 5 ⊢ (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
4		notnotr 128	. . . . . . . . . . 11 ⊢ (¬ ¬ 𝑟 = {∅} → 𝑟 = {∅})
5	4	a1i 11	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝒫 3o → (¬ ¬ 𝑟 = {∅} → 𝑟 = {∅}))
6		sssucid 6044	. . . . . . . . . . . . 13 ⊢ 2o ⊆ suc 2o
7		2on 7840	. . . . . . . . . . . . . . 15 ⊢ 2o ∈ On
8	7	elexi 3430	. . . . . . . . . . . . . 14 ⊢ 2o ∈ V
9	8	elpw 4386	. . . . . . . . . . . . 13 ⊢ (2o ∈ 𝒫 suc 2o ↔ 2o ⊆ suc 2o)
10	6, 9	mpbir 223	. . . . . . . . . . . 12 ⊢ 2o ∈ 𝒫 suc 2o
11		df2o3 7845	. . . . . . . . . . . 12 ⊢ 2o = {∅, 1o}
12		df-3o 7833	. . . . . . . . . . . . . 14 ⊢ 3o = suc 2o
13	12	eqcomi 2834	. . . . . . . . . . . . 13 ⊢ suc 2o = 3o
14	13	pweqi 4384	. . . . . . . . . . . 12 ⊢ 𝒫 suc 2o = 𝒫 3o
15	10, 11, 14	3eltr3i 2918	. . . . . . . . . . 11 ⊢ {∅, 1o} ∈ 𝒫 3o
16	15	2a1i 12	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝒫 3o → (¬ ¬ 𝑟 = {∅} → {∅, 1o} ∈ 𝒫 3o))
17	5, 16	jcad 508	. . . . . . . . 9 ⊢ (𝑟 ∈ 𝒫 3o → (¬ ¬ 𝑟 = {∅} → (𝑟 = {∅} ∧ {∅, 1o} ∈ 𝒫 3o)))
18	17	con1d 142	. . . . . . . 8 ⊢ (𝑟 ∈ 𝒫 3o → (¬ (𝑟 = {∅} ∧ {∅, 1o} ∈ 𝒫 3o) → ¬ 𝑟 = {∅}))
19	18	anc2ri 552	. . . . . . 7 ⊢ (𝑟 ∈ 𝒫 3o → (¬ (𝑟 = {∅} ∧ {∅, 1o} ∈ 𝒫 3o) → (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3o)))
20	19	orrd 894	. . . . . 6 ⊢ (𝑟 ∈ 𝒫 3o → ((𝑟 = {∅} ∧ {∅, 1o} ∈ 𝒫 3o) ∨ (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3o)))
21		ifel 4351	. . . . . 6 ⊢ (if(𝑟 = {∅}, {∅, 1o}, 𝑟) ∈ 𝒫 3o ↔ ((𝑟 = {∅} ∧ {∅, 1o} ∈ 𝒫 3o) ∨ (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3o)))
22	20, 21	sylibr 226	. . . . 5 ⊢ (𝑟 ∈ 𝒫 3o → if(𝑟 = {∅}, {∅, 1o}, 𝑟) ∈ 𝒫 3o)
23	3, 22	fmpti 6636	. . . 4 ⊢ (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)):𝒫 3o⟶𝒫 3o
24	2	pwex 5082	. . . . 5 ⊢ 𝒫 3o ∈ V
25	24, 24	elmap 8156	. . . 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 223	. . 3 ⊢ (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) ∈ (𝒫 3o ↑𝑚 𝒫 3o)
27	3	clsk1indlem0 39174	. . . . . 6 ⊢ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅
28	3	clsk1indlem2 39175	. . . . . 6 ⊢ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)
29	27, 28	pm3.2i 464	. . . . 5 ⊢ (((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠))
30	3	clsk1indlem3 39176	. . . . . 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 39177	. . . . . 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 464	. . . . 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 464	. . . 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 39178	. . . 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 464	. . 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 6436	. . . . . . . 8 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘∅) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅))
37	36	eqeq1d 2827	. . . . . . 7 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → ((𝑘‘∅) = ∅ ↔ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘∅) = ∅))
38		fveq1 6436	. . . . . . . . 9 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘𝑠) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠))
39	38	sseq2d 3858	. . . . . . . 8 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑠 ⊆ (𝑘‘𝑠) ↔ 𝑠 ⊆ ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑠)))
40	39	ralbidv 3195	. . . . . . 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 624	. . . . . 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 6436	. . . . . . . . 9 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘(𝑠 ∪ 𝑡)) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘(𝑠 ∪ 𝑡)))
43		fveq1 6436	. . . . . . . . . 10 ⊢ (𝑘 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) → (𝑘‘𝑡) = ((𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))‘𝑡))
44	38, 43	uneq12d 3997	. . . . . . . . 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 3859	. . . . . . . 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 3198	. . . . . . 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 6444	. . . . . . . . 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 2840	. . . . . . . 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 3195	. . . . . . 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 624	. . . . . 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 624	. . . . 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 3253	. . . . . 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		pm4.61 395	. . . . . . . 8 ⊢ (¬ (𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)) ↔ (𝑠 ⊆ 𝑡 ∧ ¬ (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))
55	38, 43	sseq12d 3859	. . . . . . . . . 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 310	. . . . . . . . 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 622	. . . . . . . 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 275	. . . . . . 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 3267	. . . . . 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 277	. . . . 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 624	. . . 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 3526	. . 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 683	. 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 4383	. . . . . 6 ⊢ (𝑏 = 3o → 𝒫 𝑏 = 𝒫 3o)
65	64, 64	oveq12d 6928	. . . . 5 ⊢ (𝑏 = 3o → (𝒫 𝑏 ↑𝑚 𝒫 𝑏) = (𝒫 3o ↑𝑚 𝒫 3o))
66		pm4.61 395	. . . . . 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 3356	. . . . . . . . . 10 ⊢ (𝑏 = 3o → (∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘‘𝑠) ↔ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)))
71	69, 70	syl5bb 275	. . . . . . . . 9 ⊢ (𝑏 = 3o → (𝜒 ↔ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝑘‘𝑠)))
72	68, 71	anbi12d 624	. . . . . . . 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 3356	. . . . . . . . . . 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 3364	. . . . . . . . . 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 275	. . . . . . . . 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 3356	. . . . . . . . . 10 ⊢ (𝑏 = 3o → (∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠) ↔ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠)))
79	77, 78	syl5bb 275	. . . . . . . . 9 ⊢ (𝑏 = 3o → (𝜏 ↔ ∀𝑠 ∈ 𝒫 3o(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠)))
80	76, 79	anbi12d 624	. . . . . . . 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 624	. . . . . . 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 3356	. . . . . . . . . 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 3364	. . . . . . . . 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 275	. . . . . . . 8 ⊢ (𝑏 = 3o → (𝜓 ↔ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))))
86	85	notbid 310	. . . . . . 7 ⊢ (𝑏 = 3o → (¬ 𝜓 ↔ ¬ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))))
87	81, 86	anbi12d 624	. . . . . 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 275	. . . . 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 3365	. . . 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 3526	. . 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 3253	. . . 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 3436	. . . 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 326	. . 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 210	. 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 683	1 ⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 878  ∀wal 1654   = wceq 1656   ∈ wcel 2164  ∀wral 3117  ∃wrex 3118  Vcvv 3414   ∪ cun 3796   ⊆ wss 3798  ∅c0 4146  ifcif 4308  𝒫 cpw 4380  {csn 4399  {cpr 4401   ↦ cmpt 4954  Oncon0 5967  suc csuc 5969  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910  1_oc1o 7824  2_oc2o 7825  3_oc3o 7826   ↑_𝑚 cmap 8127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-reg 8773

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-xor 1638  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-ord 5970  df-on 5971  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1o 7831  df-2o 7832  df-3o 7833  df-map 8129

This theorem is referenced by: (None)