Metamath Proof Explorer

Theorem elons

Description: Membership in the class of surreal ordinals. (Contributed by Scott Fenton, 18-Mar-2025)

Ref Expression
Assertion elons 
|- ( A e. On_s <-> ( A e. No /\ ( _Right ` A ) = (/) ) )

Proof

Step Hyp Ref Expression
1 fveq2 
 |-  ( x = A -> ( _Right ` x ) = ( _Right ` A ) )
2 1 eqeq1d 
 |-  ( x = A -> ( ( _Right ` x ) = (/) <-> ( _Right ` A ) = (/) ) )
3 df-ons 
 |-  On_s = { x e. No | ( _Right ` x ) = (/) }
4 2 3 elrab2 
 |-  ( A e. On_s <-> ( A e. No /\ ( _Right ` A ) = (/) ) )