Theorem nprmdvds1

Description: No prime number divides 1. (Contributed by Paul Chapman, 17-Nov-2012) (Proof shortened by Mario Carneiro, 2-Jul-2015)

	Ref	Expression
Assertion	nprmdvds1	⊢ ( 𝑃 ∈ ℙ → ¬ 𝑃 ∥ 1 )

Proof

Step	Hyp	Ref	Expression
1		1nprm	⊢ ¬ 1 ∈ ℙ
2		prmnn	⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℕ )
3	2	nnnn0d	⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℕ0 )
4		dvds1	⊢ ( 𝑃 ∈ ℕ0 → ( 𝑃 ∥ 1 ↔ 𝑃 = 1 ) )
5	3 4	syl	⊢ ( 𝑃 ∈ ℙ → ( 𝑃 ∥ 1 ↔ 𝑃 = 1 ) )
6		eleq1	⊢ ( 𝑃 = 1 → ( 𝑃 ∈ ℙ ↔ 1 ∈ ℙ ) )
7	6	biimpcd	⊢ ( 𝑃 ∈ ℙ → ( 𝑃 = 1 → 1 ∈ ℙ ) )
8	5 7	sylbid	⊢ ( 𝑃 ∈ ℙ → ( 𝑃 ∥ 1 → 1 ∈ ℙ ) )
9	1 8	mtoi	⊢ ( 𝑃 ∈ ℙ → ¬ 𝑃 ∥ 1 )