Theorem eluz2

Description: Membership in an upper set of integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show M e. ZZ . (Contributed by NM, 5-Sep-2005) (Revised by Mario Carneiro, 3-Nov-2013)

	Ref	Expression
Assertion	eluz2	|- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )

Proof

Step	Hyp	Ref	Expression
1		eluzel2	|- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
2		simp1	|- ( ( M e. ZZ /\ N e. ZZ /\ M <_ N ) -> M e. ZZ )
3		eluz1	|- ( M e. ZZ -> ( N e. ( ZZ>= ` M ) <-> ( N e. ZZ /\ M <_ N ) ) )
4		ibar	|- ( M e. ZZ -> ( ( N e. ZZ /\ M <_ N ) <-> ( M e. ZZ /\ ( N e. ZZ /\ M <_ N ) ) ) )
5	3 4	bitrd	|- ( M e. ZZ -> ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ ( N e. ZZ /\ M <_ N ) ) ) )
6		3anass	|- ( ( M e. ZZ /\ N e. ZZ /\ M <_ N ) <-> ( M e. ZZ /\ ( N e. ZZ /\ M <_ N ) ) )
7	5 6	bitr4di	|- ( M e. ZZ -> ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) ) )
8	1 2 7	pm5.21nii	|- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )