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Metamath Proof Explorer

Theorem slmdbn0

Description: The base set of a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018) (Proof shortened by AV, 10-Jan-2023)

Ref Expression
Hypothesis slmdbn0.b ⊢ 𝐵 = ( Base ‘ 𝑊 )
Assertion slmdbn0 ( 𝑊 ∈ SLMod → 𝐵 ≠ ∅ )

Proof

Step Hyp Ref Expression
1 slmdbn0.b ⊢ 𝐵 = ( Base ‘ 𝑊 )
2 slmdmnd ⊢ ( 𝑊 ∈ SLMod → 𝑊 ∈ Mnd )
3 1 mndbn0 ⊢ ( 𝑊 ∈ Mnd → 𝐵 ≠ ∅ )
4 2 3 syl ⊢ ( 𝑊 ∈ SLMod → 𝐵 ≠ ∅ )