Theorem elfzd

Description: Membership in a finite set of sequential integers. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	elfzd.1	$$⊢\phi \to M\in \mathbb{Z}$$
	elfzd.2	$$⊢\phi \to N\in \mathbb{Z}$$
	elfzd.3	$$⊢\phi \to K\in \mathbb{Z}$$
	elfzd.4	$$⊢\phi \to M\le K$$
	elfzd.5	$$⊢\phi \to K\le N$$
Assertion	elfzd	$$⊢\phi \to K\in \left(M\dots N\right)$$

Proof

Step	Hyp	Ref	Expression
1		elfzd.1	$$⊢\phi \to M\in \mathbb{Z}$$
2		elfzd.2	$$⊢\phi \to N\in \mathbb{Z}$$
3		elfzd.3	$$⊢\phi \to K\in \mathbb{Z}$$
4		elfzd.4	$$⊢\phi \to M\le K$$
5		elfzd.5	$$⊢\phi \to K\le N$$
6	1 2 3	3jca	$$⊢\phi \to \left(M\in \mathbb{Z}\wedge N\in \mathbb{Z}\wedge K\in \mathbb{Z}\right)$$
7	6 4 5	jca32	$$⊢\phi \to \left(\left(M\in \mathbb{Z}\wedge N\in \mathbb{Z}\wedge K\in \mathbb{Z}\right)\wedge \left(M\le K\wedge K\le N\right)\right)$$
8		elfz2	$$⊢K\in \left(M\dots N\right)\leftrightarrow \left(\left(M\in \mathbb{Z}\wedge N\in \mathbb{Z}\wedge K\in \mathbb{Z}\right)\wedge \left(M\le K\wedge K\le N\right)\right)$$
9	7 8	sylibr	$$⊢\phi \to K\in \left(M\dots N\right)$$