Metamath Proof Explorer


Theorem s1nz

Description: A singleton word is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016) (Proof shortened by Kyle Wyonch, 18-Jul-2021)

Ref Expression
Assertion s1nz ⟨“ 𝐴 ”⟩ ≠ ∅

Proof

Step Hyp Ref Expression
1 df-s1 ⟨“ 𝐴 ”⟩ = { ⟨ 0 , ( I ‘ 𝐴 ) ⟩ }
2 opex ⟨ 0 , ( I ‘ 𝐴 ) ⟩ ∈ V
3 2 snnz { ⟨ 0 , ( I ‘ 𝐴 ) ⟩ } ≠ ∅
4 1 3 eqnetri ⟨“ 𝐴 ”⟩ ≠ ∅