Theorem ceilidz

Description: A real number equals its ceiling iff it is an integer. (Contributed by AV, 30-Nov-2018)

	Ref	Expression
Assertion	ceilidz	|- ( A e. RR -> ( A e. ZZ <-> ( |^ ` A ) = A ) )

Proof

Step	Hyp	Ref	Expression
1		ceilid	|- ( A e. ZZ -> ( |^ ` A ) = A )
2		ceilcl	|- ( A e. RR -> ( |^ ` A ) e. ZZ )
3		eleq1	|- ( ( |^ ` A ) = A -> ( ( |^ ` A ) e. ZZ <-> A e. ZZ ) )
4	2 3	syl5ibcom	|- ( A e. RR -> ( ( |^ ` A ) = A -> A e. ZZ ) )
5	1 4	impbid2	|- ( A e. RR -> ( A e. ZZ <-> ( |^ ` A ) = A ) )