Metamath Proof Explorer


Theorem dfodd7

Description: Alternate definition for odd numbers. (Contributed by AV, 1-Jul-2020)

Ref Expression
Assertion dfodd7
|- Odd = { z e. ZZ | ( 2 gcd z ) = 1 }

Proof

Step Hyp Ref Expression
1 isodd7
 |-  ( x e. Odd <-> ( x e. ZZ /\ ( 2 gcd x ) = 1 ) )
2 oveq2
 |-  ( z = x -> ( 2 gcd z ) = ( 2 gcd x ) )
3 2 eqeq1d
 |-  ( z = x -> ( ( 2 gcd z ) = 1 <-> ( 2 gcd x ) = 1 ) )
4 3 elrab
 |-  ( x e. { z e. ZZ | ( 2 gcd z ) = 1 } <-> ( x e. ZZ /\ ( 2 gcd x ) = 1 ) )
5 1 4 bitr4i
 |-  ( x e. Odd <-> x e. { z e. ZZ | ( 2 gcd z ) = 1 } )
6 5 eqriv
 |-  Odd = { z e. ZZ | ( 2 gcd z ) = 1 }