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 ) |
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 ) |