Theorem dfnn2 9256

Description: Definition of the set of positive integers. Another name for df-inn 9255. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)

Assertion

Ref	Expression
dfnn2	⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Distinct variable group:   𝑥,𝑦

Proof of Theorem dfnn2

Step	Hyp	Ref	Expression
1		df-inn 9255	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 104   = wceq 1398   ∈ wcel 2205  {cab 2220  ∀wral 2522  ∩ cint 3954  (class class class)co 6058  1c1 8144   + caddc 8146  ℕcn 9254

This theorem depends on definitions:  df-inn 9255

This theorem is referenced by:  peano5nni  9257  1nn  9265  peano2nn  9266  arch  9510  caucvgre  11691