Metamath Proof Explorer

Theorem elnns

Description: Membership in the positive surreal integers. (Contributed by Scott Fenton, 15-Apr-2025)

Ref Expression
Assertion elnns 
|- ( A e. NN_s <-> ( A e. NN0_s /\ A =/= 0s ) )

Proof

Step Hyp Ref Expression
1 df-nns 
 |-  NN_s = ( NN0_s \ { 0s } )
2 1 eleq2i 
 |-  ( A e. NN_s <-> A e. ( NN0_s \ { 0s } ) )
3 eldifsn 
 |-  ( A e. ( NN0_s \ { 0s } ) <-> ( A e. NN0_s /\ A =/= 0s ) )
4 2 3 bitri 
 |-  ( A e. NN_s <-> ( A e. NN0_s /\ A =/= 0s ) )