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Metamath Proof Explorer

Theorem nn0negleid

Description: A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021)

Ref Expression
Assertion nn0negleid ( 𝐴 ∈ ℕ0 → - 𝐴 ≤ 𝐴 )

Proof

Step Hyp Ref Expression
1 nn0re ⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ )
2 1 renegcld ⊢ ( 𝐴 ∈ ℕ0 → - 𝐴 ∈ ℝ )
3 0red ⊢ ( 𝐴 ∈ ℕ0 → 0 ∈ ℝ )
4 nn0ge0 ⊢ ( 𝐴 ∈ ℕ0 → 0 ≤ 𝐴 )
5 1 le0neg2d ⊢ ( 𝐴 ∈ ℕ0 → ( 0 ≤ 𝐴 ↔ - 𝐴 ≤ 0 ) )
6 4 5 mpbid ⊢ ( 𝐴 ∈ ℕ0 → - 𝐴 ≤ 0 )
7 2 3 1 6 4 letrd ⊢ ( 𝐴 ∈ ℕ0 → - 𝐴 ≤ 𝐴 )