Theorem uzid

Description: Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005)

	Ref	Expression
Assertion	uzid	|- ( M e. ZZ -> M e. ( ZZ>= ` M ) )

Proof

Step	Hyp	Ref	Expression
1		zre	|- ( M e. ZZ -> M e. RR )
2	1	leidd	|- ( M e. ZZ -> M <_ M )
3	2	ancli	|- ( M e. ZZ -> ( M e. ZZ /\ M <_ M ) )
4		eluz1	|- ( M e. ZZ -> ( M e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ M <_ M ) ) )
5	3 4	mpbird	|- ( M e. ZZ -> M e. ( ZZ>= ` M ) )