Theorem naddov3

Description: Alternate expression for natural addition. (Contributed by Scott Fenton, 20-Jan-2025)

	Ref	Expression
Assertion	naddov3	|- ( ( A e. On /\ B e. On ) -> ( A +no B ) = |^| { x e. On | ( ( +no " ( { A } X. B ) ) u. ( +no " ( A X. { B } ) ) ) C_ x } )

Proof

Step	Hyp	Ref	Expression
1		naddov	|- ( ( A e. On /\ B e. On ) -> ( A +no B ) = |^| { x e. On | ( ( +no " ( { A } X. B ) ) C_ x /\ ( +no " ( A X. { B } ) ) C_ x ) } )
2		unss	|- ( ( ( +no " ( { A } X. B ) ) C_ x /\ ( +no " ( A X. { B } ) ) C_ x ) <-> ( ( +no " ( { A } X. B ) ) u. ( +no " ( A X. { B } ) ) ) C_ x )
3	2	rabbii	|- { x e. On | ( ( +no " ( { A } X. B ) ) C_ x /\ ( +no " ( A X. { B } ) ) C_ x ) } = { x e. On | ( ( +no " ( { A } X. B ) ) u. ( +no " ( A X. { B } ) ) ) C_ x }
4	3	inteqi	|- |^| { x e. On | ( ( +no " ( { A } X. B ) ) C_ x /\ ( +no " ( A X. { B } ) ) C_ x ) } = |^| { x e. On | ( ( +no " ( { A } X. B ) ) u. ( +no " ( A X. { B } ) ) ) C_ x }
5	1 4	eqtrdi	|- ( ( A e. On /\ B e. On ) -> ( A +no B ) = |^| { x e. On | ( ( +no " ( { A } X. B ) ) u. ( +no " ( A X. { B } ) ) ) C_ x } )