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Theorem clsk1independent 38164
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 𝑏𝑡 ∈ 𝒫 𝑏(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)))
clsnim.k2 (𝜒 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘𝑠))
clsnim.k3 (𝜃 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)))
clsnim.k4 (𝜏 ↔ ∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘𝑠)) = (𝑘𝑠))
Assertion
Ref Expression
clsk1independent ¬ ∀𝑏𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(((𝜑𝜒) ∧ (𝜃𝜏)) → 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.
StepHypRef Expression
1 3on 7555 . . 3 3𝑜 ∈ On
21elexi 3208 . 2 3𝑜 ∈ V
3 eqid 2620 . . . . 5 (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
4 notnotr 125 . . . . . . . . . . 11 (¬ ¬ 𝑟 = {∅} → 𝑟 = {∅})
54a1i 11 . . . . . . . . . 10 (𝑟 ∈ 𝒫 3𝑜 → (¬ ¬ 𝑟 = {∅} → 𝑟 = {∅}))
6 sssucid 5790 . . . . . . . . . . . . 13 2𝑜 ⊆ suc 2𝑜
7 2on 7553 . . . . . . . . . . . . . . 15 2𝑜 ∈ On
87elexi 3208 . . . . . . . . . . . . . 14 2𝑜 ∈ V
98elpw 4155 . . . . . . . . . . . . 13 (2𝑜 ∈ 𝒫 suc 2𝑜 ↔ 2𝑜 ⊆ suc 2𝑜)
106, 9mpbir 221 . . . . . . . . . . . 12 2𝑜 ∈ 𝒫 suc 2𝑜
11 df2o3 7558 . . . . . . . . . . . 12 2𝑜 = {∅, 1𝑜}
12 df-3o 7547 . . . . . . . . . . . . . 14 3𝑜 = suc 2𝑜
1312eqcomi 2629 . . . . . . . . . . . . 13 suc 2𝑜 = 3𝑜
1413pweqi 4153 . . . . . . . . . . . 12 𝒫 suc 2𝑜 = 𝒫 3𝑜
1510, 11, 143eltr3i 2711 . . . . . . . . . . 11 {∅, 1𝑜} ∈ 𝒫 3𝑜
16152a1i 12 . . . . . . . . . 10 (𝑟 ∈ 𝒫 3𝑜 → (¬ ¬ 𝑟 = {∅} → {∅, 1𝑜} ∈ 𝒫 3𝑜))
175, 16jcad 555 . . . . . . . . 9 (𝑟 ∈ 𝒫 3𝑜 → (¬ ¬ 𝑟 = {∅} → (𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜)))
1817con1d 139 . . . . . . . 8 (𝑟 ∈ 𝒫 3𝑜 → (¬ (𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜) → ¬ 𝑟 = {∅}))
1918anc2ri 580 . . . . . . 7 (𝑟 ∈ 𝒫 3𝑜 → (¬ (𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜) → (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3𝑜)))
2019orrd 393 . . . . . 6 (𝑟 ∈ 𝒫 3𝑜 → ((𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜) ∨ (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3𝑜)))
21 ifel 4120 . . . . . 6 (if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) ∈ 𝒫 3𝑜 ↔ ((𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜) ∨ (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3𝑜)))
2220, 21sylibr 224 . . . . 5 (𝑟 ∈ 𝒫 3𝑜 → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) ∈ 𝒫 3𝑜)
233, 22fmpti 6369 . . . 4 (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)):𝒫 3𝑜⟶𝒫 3𝑜
242pwex 4839 . . . . 5 𝒫 3𝑜 ∈ V
2524, 24elmap 7871 . . . 4 ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) ∈ (𝒫 3𝑜𝑚 𝒫 3𝑜) ↔ (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)):𝒫 3𝑜⟶𝒫 3𝑜)
2623, 25mpbir 221 . . 3 (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) ∈ (𝒫 3𝑜𝑚 𝒫 3𝑜)
273clsk1indlem0 38159 . . . . . 6 ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅
283clsk1indlem2 38160 . . . . . 6 𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)
2927, 28pm3.2i 471 . . . . 5 (((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠))
303clsk1indlem3 38161 . . . . . 6 𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘(𝑠𝑡)) ⊆ (((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠) ∪ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡))
313clsk1indlem4 38162 . . . . . 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)
3230, 31pm3.2i 471 . . . . 5 (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘(𝑠𝑡)) ⊆ (((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))
3329, 32pm3.2i 471 . . . 4 ((((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘(𝑠𝑡)) ⊆ (((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)))
343clsk1indlem1 38163 . . . 4 𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡))
3533, 34pm3.2i 471 . . 3 (((((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘(𝑠𝑡)) ⊆ (((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 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡)))
36 fveq1 6177 . . . . . . . 8 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑘‘∅) = ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅))
3736eqeq1d 2622 . . . . . . 7 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → ((𝑘‘∅) = ∅ ↔ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅))
38 fveq1 6177 . . . . . . . . 9 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑘𝑠) = ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠))
3938sseq2d 3625 . . . . . . . 8 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑠 ⊆ (𝑘𝑠) ↔ 𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)))
4039ralbidv 2983 . . . . . . 7 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠) ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)))
4137, 40anbi12d 746 . . . . . 6 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)) ↔ (((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠))))
42 fveq1 6177 . . . . . . . . 9 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑘‘(𝑠𝑡)) = ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘(𝑠𝑡)))
43 fveq1 6177 . . . . . . . . . 10 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑘𝑡) = ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡))
4438, 43uneq12d 3760 . . . . . . . . 9 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → ((𝑘𝑠) ∪ (𝑘𝑡)) = (((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠) ∪ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡)))
4542, 44sseq12d 3626 . . . . . . . 8 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → ((𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ↔ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘(𝑠𝑡)) ⊆ (((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠) ∪ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡))))
46452ralbidv 2986 . . . . . . 7 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘(𝑠𝑡)) ⊆ (((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)))
4847, 38fveq12d 6184 . . . . . . . . 9 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑘‘(𝑘𝑠)) = ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)))
4948, 38eqeq12d 2635 . . . . . . . 8 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → ((𝑘‘(𝑘𝑠)) = (𝑘𝑠) ↔ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)) = ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)))
5049ralbidv 2983 . . . . . . 7 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠) ↔ ∀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)))
5146, 50anbi12d 746 . . . . . 6 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → ((∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠)) ↔ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘(𝑠𝑡)) ⊆ (((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))))
5241, 51anbi12d 746 . . . . 5 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → ((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠))) ↔ ((((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘(𝑠𝑡)) ⊆ (((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 3039 . . . . . 6 (∃𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜 ¬ (𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)) ↔ ¬ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)))
54 pm4.61 442 . . . . . . . 8 (¬ (𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)) ↔ (𝑠𝑡 ∧ ¬ (𝑘𝑠) ⊆ (𝑘𝑡)))
5538, 43sseq12d 3626 . . . . . . . . . 10 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → ((𝑘𝑠) ⊆ (𝑘𝑡) ↔ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡)))
5655notbid 308 . . . . . . . . 9 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (¬ (𝑘𝑠) ⊆ (𝑘𝑡) ↔ ¬ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡)))
5756anbi2d 739 . . . . . . . 8 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → ((𝑠𝑡 ∧ ¬ (𝑘𝑠) ⊆ (𝑘𝑡)) ↔ (𝑠𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡))))
5854, 57syl5bb 272 . . . . . . 7 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (¬ (𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)) ↔ (𝑠𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡))))
59582rexbidv 3053 . . . . . 6 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (∃𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜 ¬ (𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)) ↔ ∃𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡))))
6053, 59syl5bbr 274 . . . . 5 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (¬ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)) ↔ ∃𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡))))
6152, 60anbi12d 746 . . . 4 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡))) ↔ (((((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘(𝑠𝑡)) ⊆ (((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 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡)))))
6261rspcev 3304 . . 3 (((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) ∈ (𝒫 3𝑜𝑚 𝒫 3𝑜) ∧ (((((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘(𝑠𝑡)) ⊆ (((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 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 ∧ ¬ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠) ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡)))) → ∃𝑘 ∈ (𝒫 3𝑜𝑚 𝒫 3𝑜)((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡))))
6326, 35, 62mp2an 707 . 2 𝑘 ∈ (𝒫 3𝑜𝑚 𝒫 3𝑜)((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)))
64 pweq 4152 . . . . . 6 (𝑏 = 3𝑜 → 𝒫 𝑏 = 𝒫 3𝑜)
6564, 64oveq12d 6653 . . . . 5 (𝑏 = 3𝑜 → (𝒫 𝑏𝑚 𝒫 𝑏) = (𝒫 3𝑜𝑚 𝒫 3𝑜))
66 pm4.61 442 . . . . . 6 (¬ (((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓) ↔ (((𝜑𝜒) ∧ (𝜃𝜏)) ∧ ¬ 𝜓))
67 clsnim.k0 . . . . . . . . . 10 (𝜑 ↔ (𝑘‘∅) = ∅)
6867a1i 11 . . . . . . . . 9 (𝑏 = 3𝑜 → (𝜑 ↔ (𝑘‘∅) = ∅))
69 clsnim.k2 . . . . . . . . . 10 (𝜒 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘𝑠))
7064raleqdv 3139 . . . . . . . . . 10 (𝑏 = 3𝑜 → (∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘𝑠) ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)))
7169, 70syl5bb 272 . . . . . . . . 9 (𝑏 = 3𝑜 → (𝜒 ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)))
7268, 71anbi12d 746 . . . . . . . 8 (𝑏 = 3𝑜 → ((𝜑𝜒) ↔ ((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠))))
73 clsnim.k3 . . . . . . . . . 10 (𝜃 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)))
7464raleqdv 3139 . . . . . . . . . . 11 (𝑏 = 3𝑜 → (∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ↔ ∀𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡))))
7564, 74raleqbidv 3147 . . . . . . . . . 10 (𝑏 = 3𝑜 → (∀𝑠 ∈ 𝒫 𝑏𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡))))
7673, 75syl5bb 272 . . . . . . . . 9 (𝑏 = 3𝑜 → (𝜃 ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡))))
77 clsnim.k4 . . . . . . . . . 10 (𝜏 ↔ ∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘𝑠)) = (𝑘𝑠))
7864raleqdv 3139 . . . . . . . . . 10 (𝑏 = 3𝑜 → (∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘𝑠)) = (𝑘𝑠) ↔ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠)))
7977, 78syl5bb 272 . . . . . . . . 9 (𝑏 = 3𝑜 → (𝜏 ↔ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠)))
8076, 79anbi12d 746 . . . . . . . 8 (𝑏 = 3𝑜 → ((𝜃𝜏) ↔ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠))))
8172, 80anbi12d 746 . . . . . . 7 (𝑏 = 3𝑜 → (((𝜑𝜒) ∧ (𝜃𝜏)) ↔ (((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠)))))
82 clsnim.k1 . . . . . . . . 9 (𝜓 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑡 ∈ 𝒫 𝑏(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)))
8364raleqdv 3139 . . . . . . . . . 10 (𝑏 = 3𝑜 → (∀𝑡 ∈ 𝒫 𝑏(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)) ↔ ∀𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡))))
8464, 83raleqbidv 3147 . . . . . . . . 9 (𝑏 = 3𝑜 → (∀𝑠 ∈ 𝒫 𝑏𝑡 ∈ 𝒫 𝑏(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)) ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡))))
8582, 84syl5bb 272 . . . . . . . 8 (𝑏 = 3𝑜 → (𝜓 ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡))))
8685notbid 308 . . . . . . 7 (𝑏 = 3𝑜 → (¬ 𝜓 ↔ ¬ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡))))
8781, 86anbi12d 746 . . . . . 6 (𝑏 = 3𝑜 → ((((𝜑𝜒) ∧ (𝜃𝜏)) ∧ ¬ 𝜓) ↔ ((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)))))
8866, 87syl5bb 272 . . . . 5 (𝑏 = 3𝑜 → (¬ (((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓) ↔ ((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)))))
8965, 88rexeqbidv 3148 . . . 4 (𝑏 = 3𝑜 → (∃𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ¬ (((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓) ↔ ∃𝑘 ∈ (𝒫 3𝑜𝑚 𝒫 3𝑜)((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)))))
9089rspcev 3304 . . 3 ((3𝑜 ∈ V ∧ ∃𝑘 ∈ (𝒫 3𝑜𝑚 𝒫 3𝑜)((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)))) → ∃𝑏 ∈ V ∃𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ¬ (((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓))
91 rexnal2 3039 . . . 4 (∃𝑏 ∈ V ∃𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ¬ (((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓) ↔ ¬ ∀𝑏 ∈ V ∀𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓))
92 ralv 3214 . . . 4 (∀𝑏 ∈ V ∀𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓) ↔ ∀𝑏𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓))
9391, 92xchbinx 324 . . 3 (∃𝑏 ∈ V ∃𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ¬ (((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓) ↔ ¬ ∀𝑏𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓))
9490, 93sylib 208 . 2 ((3𝑜 ∈ V ∧ ∃𝑘 ∈ (𝒫 3𝑜𝑚 𝒫 3𝑜)((((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)) ∧ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠))) ∧ ¬ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)))) → ¬ ∀𝑏𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓))
952, 63, 94mp2an 707 1 ¬ ∀𝑏𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  wal 1479   = wceq 1481  wcel 1988  wral 2909  wrex 2910  Vcvv 3195  cun 3565  wss 3567  c0 3907  ifcif 4077  𝒫 cpw 4149  {csn 4168  {cpr 4170  cmpt 4720  Oncon0 5711  suc csuc 5713  wf 5872  cfv 5876  (class class class)co 6635  1𝑜c1o 7538  2𝑜c2o 7539  3𝑜c3o 7540  𝑚 cmap 7842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-reg 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-xor 1463  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1o 7545  df-2o 7546  df-3o 7547  df-map 7844
This theorem is referenced by: (None)
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