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Subset and Equality (Continued)

Proof (continued):

• We prove (x subset y /\  y subset x) => x = y. Assume x subset y /\  y subset x, i.e., by definition of ` subset '

  (1) forall z in x: z in y;

  (2) forall z in y: z in x.

  We prove x = y, i.e., by definition of `=', forall z: z in x <=> z in y. Take arbitrary z. We have to prove z in x <=> z in y.
  • We prove z in x => z in y. Assume (3) z in x. We have to prove z in y which is a consequence of (1) and (3).
  • We prove z in y => z in x. Assume (4) z in y. We have to prove z in x which is a consequence of (2) and (4).

A well-structured argument based on definitions and given knowledge.