Theorem 1elfzo1

Description: 1 is in a half-open range of positive integers iff its upper bound is greater than 1. (Contributed by AV, 22-Nov-2022)

	Ref	Expression
Assertion	1elfzo1	|- ( 1 e. ( 1 ..^ M ) <-> ( M e. NN /\ 1 < M ) )

Proof

Step	Hyp	Ref	Expression
1		elfzo1	|- ( 1 e. ( 1 ..^ M ) <-> ( 1 e. NN /\ M e. NN /\ 1 < M ) )
2		1nn	|- 1 e. NN
3		3anass	|- ( ( 1 e. NN /\ M e. NN /\ 1 < M ) <-> ( 1 e. NN /\ ( M e. NN /\ 1 < M ) ) )
4	2 3	mpbiran	|- ( ( 1 e. NN /\ M e. NN /\ 1 < M ) <-> ( M e. NN /\ 1 < M ) )
5	1 4	bitri	|- ( 1 e. ( 1 ..^ M ) <-> ( M e. NN /\ 1 < M ) )