Theorem nnasmo

Description: There is at most one left additive inverse for natural number addition. (Contributed by Scott Fenton, 17-Oct-2024)

	Ref	Expression
Assertion	nnasmo	|- ( A e. _om -> E* x e. _om ( A +o x ) = B )

Proof

Step	Hyp	Ref	Expression
1		eqtr3	|- ( ( ( A +o x ) = B /\ ( A +o y ) = B ) -> ( A +o x ) = ( A +o y ) )
2		nnacan	|- ( ( A e. _om /\ x e. _om /\ y e. _om ) -> ( ( A +o x ) = ( A +o y ) <-> x = y ) )
3	1 2	imbitrid	|- ( ( A e. _om /\ x e. _om /\ y e. _om ) -> ( ( ( A +o x ) = B /\ ( A +o y ) = B ) -> x = y ) )
4	3	3expb	|- ( ( A e. _om /\ ( x e. _om /\ y e. _om ) ) -> ( ( ( A +o x ) = B /\ ( A +o y ) = B ) -> x = y ) )
5	4	ralrimivva	|- ( A e. _om -> A. x e. _om A. y e. _om ( ( ( A +o x ) = B /\ ( A +o y ) = B ) -> x = y ) )
6		oveq2	|- ( x = y -> ( A +o x ) = ( A +o y ) )
7	6	eqeq1d	|- ( x = y -> ( ( A +o x ) = B <-> ( A +o y ) = B ) )
8	7	rmo4	|- ( E* x e. _om ( A +o x ) = B <-> A. x e. _om A. y e. _om ( ( ( A +o x ) = B /\ ( A +o y ) = B ) -> x = y ) )
9	5 8	sylibr	|- ( A e. _om -> E* x e. _om ( A +o x ) = B )