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 ∈ Odd ↔ ( 𝑍 ∈ ℤ ∧ ¬ 2 ∥ 𝑍 ) )

Proof

Step	Hyp	Ref	Expression
1		breq2	⊢ ( 𝑧 = 𝑍 → ( 2 ∥ 𝑧 ↔ 2 ∥ 𝑍 ) )
2	1	notbid	⊢ ( 𝑧 = 𝑍 → ( ¬ 2 ∥ 𝑧 ↔ ¬ 2 ∥ 𝑍 ) )
3		dfodd3	⊢ Odd = { 𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧 }
4	2 3	elrab2	⊢ ( 𝑍 ∈ Odd ↔ ( 𝑍 ∈ ℤ ∧ ¬ 2 ∥ 𝑍 ) )