Theorem dfnn2 9020

Description: Definition of the set of positive integers. Another name for df-inn 9019. (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 9019	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 104   = wceq 1372   ∈ wcel 2175  {cab 2190  ∀wral 2483  ∩ cint 3884  (class class class)co 5934  1c1 7908   + caddc 7910  ℕcn 9018

This theorem depends on definitions:  df-inn 9019

This theorem is referenced by:  peano5nni  9021  1nn  9029  peano2nn  9030  arch  9274  caucvgre  11211