Subset and Equality
Proposition: For every x and y, we have
x = y <=> (x subset y /\ y
subset x).
Proof:
Take arbitrary x and y.
We prove x = y <=> (x subset y
/\ y subset x).
- We prove x = y => (x subset y
/\ y subset x). Assume x = y, i.e., by
definition of `=',
(1) forall z: z in x <=> z in y.
We have to prove x subset y /\ y subset x.
- We prove x subset y, i.e., by definition of
` subset ',
forall z in x: z in y.
Take arbitrary z.
We have to prove z in x => z in y. Assume (2) z in x. We have to prove z
in y which is a consequence of (1) and (2).
- The proof of y subset x proceeds analogously.
Author: Wolfgang Schreiner
Last Modification: October 14, 1999