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

Metamath Proof Explorer

Theorem eluzp1p1

Description: Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005)

Ref Expression
Assertion eluzp1p1 ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) )

Proof

Step Hyp Ref Expression
1 peano2z ⊢ ( 𝑀 ∈ ℤ → ( 𝑀 + 1 ) ∈ ℤ )
2 1 3ad2ant1 ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) → ( 𝑀 + 1 ) ∈ ℤ )
3 peano2z ⊢ ( 𝑁 ∈ ℤ → ( 𝑁 + 1 ) ∈ ℤ )
4 3 3ad2ant2 ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) → ( 𝑁 + 1 ) ∈ ℤ )
5 zre ⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℝ )
6 zre ⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℝ )
7 1re ⊢ 1 ∈ ℝ
8 leadd1 ⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 1 ∈ ℝ ) → ( 𝑀 ≤ 𝑁 ↔ ( 𝑀 + 1 ) ≤ ( 𝑁 + 1 ) ) )
9 7 8 mp3an3 ⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( 𝑀 ≤ 𝑁 ↔ ( 𝑀 + 1 ) ≤ ( 𝑁 + 1 ) ) )
10 5 6 9 syl2an ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ≤ 𝑁 ↔ ( 𝑀 + 1 ) ≤ ( 𝑁 + 1 ) ) )
11 10 biimp3a ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) → ( 𝑀 + 1 ) ≤ ( 𝑁 + 1 ) )
12 2 4 11 3jca ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) → ( ( 𝑀 + 1 ) ∈ ℤ ∧ ( 𝑁 + 1 ) ∈ ℤ ∧ ( 𝑀 + 1 ) ≤ ( 𝑁 + 1 ) ) )
13 eluz2 ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) )
14 eluz2 ⊢ ( ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ↔ ( ( 𝑀 + 1 ) ∈ ℤ ∧ ( 𝑁 + 1 ) ∈ ℤ ∧ ( 𝑀 + 1 ) ≤ ( 𝑁 + 1 ) ) )
15 12 13 14 3imtr4i ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) )