Theorem isodd

Description: The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd integer increased by 1 and then divided by 2 is still an integer. (Contributed by AV, 14-Jun-2020)

	Ref	Expression
Assertion	isodd	|- ( Z e. Odd <-> ( Z e. ZZ /\ ( ( Z + 1 ) / 2 ) e. ZZ ) )

Proof

Step	Hyp	Ref	Expression
1		oveq1	|- ( z = Z -> ( z + 1 ) = ( Z + 1 ) )
2	1	oveq1d	|- ( z = Z -> ( ( z + 1 ) / 2 ) = ( ( Z + 1 ) / 2 ) )
3	2	eleq1d	|- ( z = Z -> ( ( ( z + 1 ) / 2 ) e. ZZ <-> ( ( Z + 1 ) / 2 ) e. ZZ ) )
4		df-odd	|- Odd = { z e. ZZ | ( ( z + 1 ) / 2 ) e. ZZ }
5	3 4	elrab2	|- ( Z e. Odd <-> ( Z e. ZZ /\ ( ( Z + 1 ) / 2 ) e. ZZ ) )