This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem eluzp1

Description: Membership in a successor upper set of integers. (Contributed by SN, 5-Jul-2025)

Ref Expression
Assertion eluzp1 ( 𝑀 ∈ ℤ → ( 𝑁 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↔ ( 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) ) )

Proof

Step Hyp Ref Expression
1 zltp1le ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 < 𝑁 ↔ ( 𝑀 + 1 ) ≤ 𝑁 ) )
2 1 pm5.32da ⊢ ( 𝑀 ∈ ℤ → ( ( 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) ↔ ( 𝑁 ∈ ℤ ∧ ( 𝑀 + 1 ) ≤ 𝑁 ) ) )
3 peano2z ⊢ ( 𝑀 ∈ ℤ → ( 𝑀 + 1 ) ∈ ℤ )
4 3 3biant1d ⊢ ( 𝑀 ∈ ℤ → ( ( 𝑁 ∈ ℤ ∧ ( 𝑀 + 1 ) ≤ 𝑁 ) ↔ ( ( 𝑀 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ( 𝑀 + 1 ) ≤ 𝑁 ) ) )
5 eluz2 ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↔ ( ( 𝑀 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ( 𝑀 + 1 ) ≤ 𝑁 ) )
6 4 5 bitr4di ⊢ ( 𝑀 ∈ ℤ → ( ( 𝑁 ∈ ℤ ∧ ( 𝑀 + 1 ) ≤ 𝑁 ) ↔ 𝑁 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) )
7 2 6 bitr2d ⊢ ( 𝑀 ∈ ℤ → ( 𝑁 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↔ ( 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) ) )