Theorem eluzd

Description: Membership in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	eluzd.1	|- Z = ( ZZ>= ` M )
	eluzd.2	|- ( ph -> M e. ZZ )
	eluzd.3	|- ( ph -> N e. ZZ )
	eluzd.4	|- ( ph -> M <_ N )
Assertion	eluzd	|- ( ph -> N e. Z )

Proof

Step	Hyp	Ref	Expression
1		eluzd.1	|- Z = ( ZZ>= ` M )
2		eluzd.2	|- ( ph -> M e. ZZ )
3		eluzd.3	|- ( ph -> N e. ZZ )
4		eluzd.4	|- ( ph -> M <_ N )
5		eluz2	|- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
6	2 3 4 5	syl3anbrc	|- ( ph -> N e. ( ZZ>= ` M ) )
7	6 1	eleqtrrdi	|- ( ph -> N e. Z )