Theorem isodd3

Description: The predicate "is an odd number". An odd number is an integer which is not divisible by 2. (Contributed by AV, 18-Jun-2020)

	Ref	Expression
Assertion	isodd3	$$⊢Z\in \mathrm{Odd}\leftrightarrow \left(Z\in \mathbb{Z}\wedge \neg 2\parallel Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		breq2	$$⊢z=Z\to \left(2\parallel z\leftrightarrow 2\parallel Z\right)$$
2	1	notbid	$$⊢z=Z\to \left(\neg 2\parallel z\leftrightarrow \neg 2\parallel Z\right)$$
3		dfodd3	$$⊢\mathrm{Odd}=\left\{z\in \mathbb{Z}|\neg 2\parallel z\right\}$$
4	2 3	elrab2	$$⊢Z\in \mathrm{Odd}\leftrightarrow \left(Z\in \mathbb{Z}\wedge \neg 2\parallel Z\right)$$