Theorem 1nprm

Description: 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011) (Proof shortened by Fan Zheng, 3-Jul-2016)

	Ref	Expression
Assertion	1nprm	⊢ ¬ 1 ∈ ℙ

Proof

Step	Hyp	Ref	Expression
1		1nn	⊢ 1 ∈ ℕ
2		eleq1	⊢ ( 𝑧 = 1 → ( 𝑧 ∈ ℕ ↔ 1 ∈ ℕ ) )
3	1 2	mpbiri	⊢ ( 𝑧 = 1 → 𝑧 ∈ ℕ )
4		nnnn0	⊢ ( 𝑧 ∈ ℕ → 𝑧 ∈ ℕ0 )
5		dvds1	⊢ ( 𝑧 ∈ ℕ0 → ( 𝑧 ∥ 1 ↔ 𝑧 = 1 ) )
6	4 5	syl	⊢ ( 𝑧 ∈ ℕ → ( 𝑧 ∥ 1 ↔ 𝑧 = 1 ) )
7	6	bicomd	⊢ ( 𝑧 ∈ ℕ → ( 𝑧 = 1 ↔ 𝑧 ∥ 1 ) )
8	3 7	biadanii	⊢ ( 𝑧 = 1 ↔ ( 𝑧 ∈ ℕ ∧ 𝑧 ∥ 1 ) )
9		velsn	⊢ ( 𝑧 ∈ { 1 } ↔ 𝑧 = 1 )
10		breq1	⊢ ( 𝑛 = 𝑧 → ( 𝑛 ∥ 1 ↔ 𝑧 ∥ 1 ) )
11	10	elrab	⊢ ( 𝑧 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 1 } ↔ ( 𝑧 ∈ ℕ ∧ 𝑧 ∥ 1 ) )
12	8 9 11	3bitr4ri	⊢ ( 𝑧 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 1 } ↔ 𝑧 ∈ { 1 } )
13	12	eqriv	⊢ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 1 } = { 1 }
14		1ex	⊢ 1 ∈ V
15	14	ensn1	⊢ { 1 } ≈ 1o
16	13 15	eqbrtri	⊢ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 1 } ≈ 1o
17		1sdom2	⊢ 1o ≺ 2o
18		ensdomtr	⊢ ( ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 1 } ≈ 1o ∧ 1o ≺ 2o ) → { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 1 } ≺ 2o )
19	16 17 18	mp2an	⊢ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 1 } ≺ 2o
20		sdomnen	⊢ ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 1 } ≺ 2o → ¬ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 1 } ≈ 2o )
21	19 20	ax-mp	⊢ ¬ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 1 } ≈ 2o
22		isprm	⊢ ( 1 ∈ ℙ ↔ ( 1 ∈ ℕ ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 1 } ≈ 2o ) )
23	1 22	mpbiran	⊢ ( 1 ∈ ℙ ↔ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 1 } ≈ 2o )
24	21 23	mtbir	⊢ ¬ 1 ∈ ℙ