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	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin ) → ( ♯ ‘ 𝐴 ) ∉ ℕ0 )

Proof

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