Metamath Proof Explorer


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 ) )