Let A be an object. Then we say A is well-ordered by membership if "A is strictly totally ordered by membership"∧∀X∈℘(A)⋅("X is nonempty"⊃∃x∈X⋅∀Y∈X⋅(x=Y∨x∈Y)).
The following background is necessary to understand this definition.
(.) Let A and x be objects. Then x∈A is a proposition.
(.) Let x and y be objects. Then x=y is a proposition.
(.) Let φ and ψ be propositions. Then φ⊃ψ is a proposition.
(.) Let φ and ψ be propositions. Then φ∧ψ is a proposition.
(.) Let φ and ψ be propositions. Then φ∨ψ is a proposition.
(.) Let A be an object. Then ℘(A) is an object.
(.) Let A be an object. Let φ(a) be a proposition depending on a member a of A. Then "∀a∈A⋅φ(a)" is a proposition.
(.) Let A be an object. Let φ(a) be a proposition depending on a member a of A. Then "∃a∈A⋅φ(a)" is a proposition.
(.) Let x be an object. Then "x is nonempty" is a proposition.
(.) Let x be an object. Then "x is strictly totally ordered by membership" is a proposition.
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