Metamath Proof Explorer


Theorem oddprmuzge3

Description: A prime number which is odd is an integer greater than or equal to 3. (Contributed by AV, 20-Jul-2020) (Proof shortened by AV, 21-Aug-2021)

Ref Expression
Assertion oddprmuzge3 ( ( 𝑃 ∈ ℙ ∧ 𝑃 ∈ Odd ) → 𝑃 ∈ ( ℤ ‘ 3 ) )

Proof

Step Hyp Ref Expression
1 oddprmne2 ( ( 𝑃 ∈ ℙ ∧ 𝑃 ∈ Odd ) ↔ 𝑃 ∈ ( ℙ ∖ { 2 } ) )
2 oddprmge3 ( 𝑃 ∈ ( ℙ ∖ { 2 } ) → 𝑃 ∈ ( ℤ ‘ 3 ) )
3 1 2 sylbi ( ( 𝑃 ∈ ℙ ∧ 𝑃 ∈ Odd ) → 𝑃 ∈ ( ℤ ‘ 3 ) )