Theorem nnuzdisj

Description: The first N elements of the set of nonnegative integers are distinct from any later members. (Contributed by Glauco Siliprandi, 21-Nov-2020)

	Ref	Expression
Assertion	nnuzdisj	$$⊢\left(1\dots N\right)\cap {\mathbb{Z}}_{\ge \left(N+1\right)}=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		fz1ssfz0	$$⊢\left(1\dots N\right)\subseteq \left(0\dots N\right)$$
2		ssrin	$$⊢\left(1\dots N\right)\subseteq \left(0\dots N\right)\to \left(1\dots N\right)\cap {\mathbb{Z}}_{\ge \left(N+1\right)}\subseteq \left(0\dots N\right)\cap {\mathbb{Z}}_{\ge \left(N+1\right)}$$
3	1 2	ax-mp	$$⊢\left(1\dots N\right)\cap {\mathbb{Z}}_{\ge \left(N+1\right)}\subseteq \left(0\dots N\right)\cap {\mathbb{Z}}_{\ge \left(N+1\right)}$$
4		nn0disj	$$⊢\left(0\dots N\right)\cap {\mathbb{Z}}_{\ge \left(N+1\right)}=\varnothing$$
5		sseq0	$$⊢\left(\left(1\dots N\right)\cap {\mathbb{Z}}_{\ge \left(N+1\right)}\subseteq \left(0\dots N\right)\cap {\mathbb{Z}}_{\ge \left(N+1\right)}\wedge \left(0\dots N\right)\cap {\mathbb{Z}}_{\ge \left(N+1\right)}=\varnothing \right)\to \left(1\dots N\right)\cap {\mathbb{Z}}_{\ge \left(N+1\right)}=\varnothing$$
6	3 4 5	mp2an	$$⊢\left(1\dots N\right)\cap {\mathbb{Z}}_{\ge \left(N+1\right)}=\varnothing$$