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Theorem clsk1independent 37364
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 7457 . . 3 3𝑜 ∈ On
21elexi 3186 . 2 3𝑜 ∈ V
3 eqid 2610 . . . . 5 (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))
4 notnotr 124 . . . . . . . . . . 11 (¬ ¬ 𝑟 = {∅} → 𝑟 = {∅})
54a1i 11 . . . . . . . . . 10 (𝑟 ∈ 𝒫 3𝑜 → (¬ ¬ 𝑟 = {∅} → 𝑟 = {∅}))
6 sssucid 5719 . . . . . . . . . . . . 13 2𝑜 ⊆ suc 2𝑜
7 2on 7455 . . . . . . . . . . . . . . 15 2𝑜 ∈ On
87elexi 3186 . . . . . . . . . . . . . 14 2𝑜 ∈ V
98elpw 4114 . . . . . . . . . . . . 13 (2𝑜 ∈ 𝒫 suc 2𝑜 ↔ 2𝑜 ⊆ suc 2𝑜)
106, 9mpbir 220 . . . . . . . . . . . 12 2𝑜 ∈ 𝒫 suc 2𝑜
11 df2o3 7460 . . . . . . . . . . . 12 2𝑜 = {∅, 1𝑜}
12 df-3o 7449 . . . . . . . . . . . . . 14 3𝑜 = suc 2𝑜
1312eqcomi 2619 . . . . . . . . . . . . 13 suc 2𝑜 = 3𝑜
1413pweqi 4112 . . . . . . . . . . . 12 𝒫 suc 2𝑜 = 𝒫 3𝑜
1510, 11, 143eltr3i 2700 . . . . . . . . . . 11 {∅, 1𝑜} ∈ 𝒫 3𝑜
16152a1i 12 . . . . . . . . . 10 (𝑟 ∈ 𝒫 3𝑜 → (¬ ¬ 𝑟 = {∅} → {∅, 1𝑜} ∈ 𝒫 3𝑜))
175, 16jcad 554 . . . . . . . . 9 (𝑟 ∈ 𝒫 3𝑜 → (¬ ¬ 𝑟 = {∅} → (𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜)))
1817con1d 138 . . . . . . . 8 (𝑟 ∈ 𝒫 3𝑜 → (¬ (𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜) → ¬ 𝑟 = {∅}))
1918anc2ri 579 . . . . . . 7 (𝑟 ∈ 𝒫 3𝑜 → (¬ (𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜) → (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3𝑜)))
2019orrd 392 . . . . . 6 (𝑟 ∈ 𝒫 3𝑜 → ((𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜) ∨ (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3𝑜)))
21 ifel 4079 . . . . . 6 (if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) ∈ 𝒫 3𝑜 ↔ ((𝑟 = {∅} ∧ {∅, 1𝑜} ∈ 𝒫 3𝑜) ∨ (¬ 𝑟 = {∅} ∧ 𝑟 ∈ 𝒫 3𝑜)))
2220, 21sylibr 223 . . . . 5 (𝑟 ∈ 𝒫 3𝑜 → if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟) ∈ 𝒫 3𝑜)
233, 22fmpti 6291 . . . 4 (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)):𝒫 3𝑜⟶𝒫 3𝑜
242pwex 4774 . . . . 5 𝒫 3𝑜 ∈ V
2524, 24elmap 7772 . . . 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 220 . . 3 (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) ∈ (𝒫 3𝑜𝑚 𝒫 3𝑜)
273clsk1indlem0 37359 . . . . . 6 ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅
283clsk1indlem2 37360 . . . . . 6 𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)
2927, 28pm3.2i 470 . . . . 5 (((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠))
303clsk1indlem3 37361 . . . . . 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 37362 . . . . . 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 470 . . . . 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 470 . . . 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 37363 . . . 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 470 . . 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 6102 . . . . . . . 8 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑘‘∅) = ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅))
3736eqeq1d 2612 . . . . . . 7 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → ((𝑘‘∅) = ∅ ↔ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘∅) = ∅))
38 fveq1 6102 . . . . . . . . 9 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑘𝑠) = ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠))
3938sseq2d 3596 . . . . . . . 8 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑠 ⊆ (𝑘𝑠) ↔ 𝑠 ⊆ ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑠)))
4039ralbidv 2969 . . . . . . 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 743 . . . . . 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 6102 . . . . . . . . 9 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑘‘(𝑠𝑡)) = ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘(𝑠𝑡)))
43 fveq1 6102 . . . . . . . . . 10 (𝑘 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) → (𝑘𝑡) = ((𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟))‘𝑡))
4438, 43uneq12d 3730 . . . . . . . . 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 3597 . . . . . . . 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 2972 . . . . . . 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 6109 . . . . . . . . 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 2625 . . . . . . . 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 2969 . . . . . . 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 743 . . . . . 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 743 . . . . 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 3025 . . . . . 6 (∃𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜 ¬ (𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)) ↔ ¬ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)))
54 pm4.61 441 . . . . . . . 8 (¬ (𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)) ↔ (𝑠𝑡 ∧ ¬ (𝑘𝑠) ⊆ (𝑘𝑡)))
5538, 43sseq12d 3597 . . . . . . . . . 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 307 . . . . . . . . 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 736 . . . . . . . 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 271 . . . . . . 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 3039 . . . . . 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 273 . . . . 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 743 . . . 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 3282 . . 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 704 . 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 4111 . . . . . 6 (𝑏 = 3𝑜 → 𝒫 𝑏 = 𝒫 3𝑜)
6564, 64oveq12d 6567 . . . . 5 (𝑏 = 3𝑜 → (𝒫 𝑏𝑚 𝒫 𝑏) = (𝒫 3𝑜𝑚 𝒫 3𝑜))
66 pm4.61 441 . . . . . 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 3121 . . . . . . . . . 10 (𝑏 = 3𝑜 → (∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘𝑠) ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)))
7169, 70syl5bb 271 . . . . . . . . 9 (𝑏 = 3𝑜 → (𝜒 ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠)))
7268, 71anbi12d 743 . . . . . . . 8 (𝑏 = 3𝑜 → ((𝜑𝜒) ↔ ((𝑘‘∅) = ∅ ∧ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝑘𝑠))))
73 clsnim.k3 . . . . . . . . . 10 (𝜃 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)))
7464raleqdv 3121 . . . . . . . . . . 11 (𝑏 = 3𝑜 → (∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ↔ ∀𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡))))
7564, 74raleqbidv 3129 . . . . . . . . . 10 (𝑏 = 3𝑜 → (∀𝑠 ∈ 𝒫 𝑏𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡))))
7673, 75syl5bb 271 . . . . . . . . 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 3121 . . . . . . . . . 10 (𝑏 = 3𝑜 → (∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘𝑠)) = (𝑘𝑠) ↔ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠)))
7977, 78syl5bb 271 . . . . . . . . 9 (𝑏 = 3𝑜 → (𝜏 ↔ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠)))
8076, 79anbi12d 743 . . . . . . . 8 (𝑏 = 3𝑜 → ((𝜃𝜏) ↔ (∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑘‘(𝑠𝑡)) ⊆ ((𝑘𝑠) ∪ (𝑘𝑡)) ∧ ∀𝑠 ∈ 𝒫 3𝑜(𝑘‘(𝑘𝑠)) = (𝑘𝑠))))
8172, 80anbi12d 743 . . . . . . 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 3121 . . . . . . . . . 10 (𝑏 = 3𝑜 → (∀𝑡 ∈ 𝒫 𝑏(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)) ↔ ∀𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡))))
8464, 83raleqbidv 3129 . . . . . . . . 9 (𝑏 = 3𝑜 → (∀𝑠 ∈ 𝒫 𝑏𝑡 ∈ 𝒫 𝑏(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡)) ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡))))
8582, 84syl5bb 271 . . . . . . . 8 (𝑏 = 3𝑜 → (𝜓 ↔ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡))))
8685notbid 307 . . . . . . 7 (𝑏 = 3𝑜 → (¬ 𝜓 ↔ ¬ ∀𝑠 ∈ 𝒫 3𝑜𝑡 ∈ 𝒫 3𝑜(𝑠𝑡 → (𝑘𝑠) ⊆ (𝑘𝑡))))
8781, 86anbi12d 743 . . . . . 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 271 . . . . 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 3130 . . . 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 3282 . . 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 3025 . . . 4 (∃𝑏 ∈ V ∃𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ¬ (((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓) ↔ ¬ ∀𝑏 ∈ V ∀𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓))
92 ralv 3192 . . . 4 (∀𝑏 ∈ V ∀𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓) ↔ ∀𝑏𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓))
9391, 92xchbinx 323 . . 3 (∃𝑏 ∈ V ∃𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ¬ (((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓) ↔ ¬ ∀𝑏𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓))
9490, 93sylib 207 . 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 704 1 ¬ ∀𝑏𝑘 ∈ (𝒫 𝑏𝑚 𝒫 𝑏)(((𝜑𝜒) ∧ (𝜃𝜏)) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383  wal 1473   = wceq 1475  wcel 1977  wral 2896  wrex 2897  Vcvv 3173  cun 3538  wss 3540  c0 3874  ifcif 4036  𝒫 cpw 4108  {csn 4125  {cpr 4127  cmpt 4643  Oncon0 5640  suc csuc 5642  wf 5800  cfv 5804  (class class class)co 6549  1𝑜c1o 7440  2𝑜c2o 7441  3𝑜c3o 7442  𝑚 cmap 7744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-xor 1457  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1o 7447  df-2o 7448  df-3o 7449  df-map 7746
This theorem is referenced by: (None)
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