Metamath Proof Explorer


Theorem hashnfinnn0

Description: The size of an infinite set is not a nonnegative integer. (Contributed by Alexander van der Vekens, 21-Dec-2017) (Proof shortened by Alexander van der Vekens, 18-Jan-2018)

Ref Expression
Assertion hashnfinnn0 ( ( 𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ( ♯ ‘ 𝐴 ) ∉ ℕ0 )

Proof

Step Hyp Ref Expression
1 nnel ( ¬ ( ♯ ‘ 𝐴 ) ∉ ℕ0 ↔ ( ♯ ‘ 𝐴 ) ∈ ℕ0 )
2 hashclb ( 𝐴𝑉 → ( 𝐴 ∈ Fin ↔ ( ♯ ‘ 𝐴 ) ∈ ℕ0 ) )
3 2 biimprd ( 𝐴𝑉 → ( ( ♯ ‘ 𝐴 ) ∈ ℕ0𝐴 ∈ Fin ) )
4 1 3 syl5bi ( 𝐴𝑉 → ( ¬ ( ♯ ‘ 𝐴 ) ∉ ℕ0𝐴 ∈ Fin ) )
5 4 con1d ( 𝐴𝑉 → ( ¬ 𝐴 ∈ Fin → ( ♯ ‘ 𝐴 ) ∉ ℕ0 ) )
6 5 imp ( ( 𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ( ♯ ‘ 𝐴 ) ∉ ℕ0 )