Theorem elfz1uz

Description: Membership in a 1-based finite set of sequential integers with an upper integer. (Contributed by AV, 23-Jan-2022)

	Ref	Expression
Assertion	elfz1uz	|- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> N e. ( 1 ... M ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	|- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> N e. NN )
2		eluzle	|- ( M e. ( ZZ>= ` N ) -> N <_ M )
3	2	adantl	|- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> N <_ M )
4		eluzelz	|- ( M e. ( ZZ>= ` N ) -> M e. ZZ )
5	4	adantl	|- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> M e. ZZ )
6		fznn	|- ( M e. ZZ -> ( N e. ( 1 ... M ) <-> ( N e. NN /\ N <_ M ) ) )
7	5 6	syl	|- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> ( N e. ( 1 ... M ) <-> ( N e. NN /\ N <_ M ) ) )
8	1 3 7	mpbir2and	|- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> N e. ( 1 ... M ) )