Theorem peano2n0s

Description: Peano postulate: the successor of a non-negative surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025)

	Ref	Expression
Assertion	peano2n0s	|- ( A e. NN0_s -> ( A +s 1s ) e. NN0_s )

Proof

Step	Hyp	Ref	Expression
1		df-n0s	|- NN0_s = ( rec ( ( x e. _V |-> ( x +s 1s ) ) , 0s ) " _om )
2	1	a1i	|- ( A e. NN0_s -> NN0_s = ( rec ( ( x e. _V |-> ( x +s 1s ) ) , 0s ) " _om ) )
3		0sno	|- 0s e. No
4	3	a1i	|- ( A e. NN0_s -> 0s e. No )
5		id	|- ( A e. NN0_s -> A e. NN0_s )
6	2 4 5	noseqp1	|- ( A e. NN0_s -> ( A +s 1s ) e. NN0_s )