Now consider a finite set A and some element b not in A. then by definition of finite sets the inductive-A-system is P(A). Let B = A ∪ {b} and let R be an inductive-B-system. Thus for all C⊆R we have (∅ in C and X in C ⟹ X ∪ {b} in C for x in B) implies B in C

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In particular, B= A ∪ {b} in C suggests A in C and hence P(A) in C by the definition of inductive-A-system. Moreover, {b} in C and by the definition of inductive-B-system, {b}∪ D in C, where D in P(A)

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In particular, B= A ∪ {b} in C suggests A in C and hence D in C, where D in P(A) by the definition of inductive-A-system. Moreover, {b} in C and by the definition of inductive-B-system, {b}∪ D in C

Hence P(A) ⊆ R

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For deletion: Z. Y. X.

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