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Metamath Proof Explorer

Theorem btwnnz

Description: A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005)

Ref Expression
Assertion btwnnz ( ( 𝐴 ∈ ℤ ∧ 𝐴 < 𝐵 ∧ 𝐵 < ( 𝐴 + 1 ) ) → ¬ 𝐵 ∈ ℤ )

Proof

Step Hyp Ref Expression
1 zltp1le ⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 + 1 ) ≤ 𝐵 ) )
2 peano2z ⊢ ( 𝐴 ∈ ℤ → ( 𝐴 + 1 ) ∈ ℤ )
3 zre ⊢ ( ( 𝐴 + 1 ) ∈ ℤ → ( 𝐴 + 1 ) ∈ ℝ )
4 2 3 syl ⊢ ( 𝐴 ∈ ℤ → ( 𝐴 + 1 ) ∈ ℝ )
5 zre ⊢ ( 𝐵 ∈ ℤ → 𝐵 ∈ ℝ )
6 lenlt ⊢ ( ( ( 𝐴 + 1 ) ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 + 1 ) ≤ 𝐵 ↔ ¬ 𝐵 < ( 𝐴 + 1 ) ) )
7 4 5 6 syl2an ⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 + 1 ) ≤ 𝐵 ↔ ¬ 𝐵 < ( 𝐴 + 1 ) ) )
8 1 7 bitrd ⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 < 𝐵 ↔ ¬ 𝐵 < ( 𝐴 + 1 ) ) )
9 8 biimpd ⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 < 𝐵 → ¬ 𝐵 < ( 𝐴 + 1 ) ) )
10 9 impancom ⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐴 < 𝐵 ) → ( 𝐵 ∈ ℤ → ¬ 𝐵 < ( 𝐴 + 1 ) ) )
11 10 con2d ⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐴 < 𝐵 ) → ( 𝐵 < ( 𝐴 + 1 ) → ¬ 𝐵 ∈ ℤ ) )
12 11 3impia ⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐴 < 𝐵 ∧ 𝐵 < ( 𝐴 + 1 ) ) → ¬ 𝐵 ∈ ℤ )