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 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		eluzel2	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ )
2		simp1	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) → 𝑀 ∈ ℤ )
3		eluz1	⊢ ( 𝑀 ∈ ℤ → ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) ) )
4		ibar	⊢ ( 𝑀 ∈ ℤ → ( ( 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) ↔ ( 𝑀 ∈ ℤ ∧ ( 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) ) ) )
5	3 4	bitrd	⊢ ( 𝑀 ∈ ℤ → ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ ( 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) ) ) )
6		3anass	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) ↔ ( 𝑀 ∈ ℤ ∧ ( 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) ) )
7	5 6	bitr4di	⊢ ( 𝑀 ∈ ℤ → ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) ) )
8	1 2 7	pm5.21nii	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) )