Metamath Proof Explorer


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 ∈ ( ℤ𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁 ) )

Proof

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