-interpretation "subset construction" 'subset \eta.x = (mk_powerset _ x).
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-definition mem ≝ λA.λS:Ω \sup A.λx:A. match S with [mk_powerset c ⇒ c x].
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-interpretation "mem" 'mem a S = (mem _ S a).
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-definition overlaps ≝ λA:Type.λU,V:Ω \sup A.exT2 ? (λa:A. a ∈ U) (λa.a ∈ V).
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-interpretation "overlaps" 'overlaps U V = (overlaps _ U V).
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-definition subseteq ≝ λA:Type.λU,V:Ω \sup A.∀a:A. a ∈ U → a ∈ V.
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-interpretation "subseteq" 'subseteq U V = (subseteq _ U V).
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-definition intersects ≝ λA:Type.λU,V:Ω \sup A.{a | a ∈ U ∧ a ∈ V}.
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-interpretation "intersects" 'intersects U V = (intersects _ U V).
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-definition union ≝ λA:Type.λU,V:Ω \sup A.{a | a ∈ U ∨ a ∈ V}.
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-interpretation "union" 'union U V = (union _ U V).