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Metamath Proof Explorer

Theorem oddprmne2

Description: Every prime number not being 2 is an odd prime number. (Contributed by AV, 21-Aug-2021)

Ref Expression
Assertion oddprmne2 
|- ( ( P e. Prime /\ P e. Odd ) <-> P e. ( Prime \ { 2 } ) )

Proof

Step Hyp Ref Expression
1 prmz 
 |-  ( P e. Prime -> P e. ZZ )
2 zeo2ALTV 
 |-  ( P e. ZZ -> ( P e. Even <-> -. P e. Odd ) )
3 1 2 syl 
 |-  ( P e. Prime -> ( P e. Even <-> -. P e. Odd ) )
4 evenprm2 
 |-  ( P e. Prime -> ( P e. Even <-> P = 2 ) )
5 3 4 bitr3d 
 |-  ( P e. Prime -> ( -. P e. Odd <-> P = 2 ) )
6 nne 
 |-  ( -. P =/= 2 <-> P = 2 )
7 5 6 bitr4di 
 |-  ( P e. Prime -> ( -. P e. Odd <-> -. P =/= 2 ) )
8 7 con4bid 
 |-  ( P e. Prime -> ( P e. Odd <-> P =/= 2 ) )
9 8 pm5.32i 
 |-  ( ( P e. Prime /\ P e. Odd ) <-> ( P e. Prime /\ P =/= 2 ) )
10 eldifsn 
 |-  ( P e. ( Prime \ { 2 } ) <-> ( P e. Prime /\ P =/= 2 ) )
11 9 10 bitr4i 
 |-  ( ( P e. Prime /\ P e. Odd ) <-> P e. ( Prime \ { 2 } ) )