Theorem dfnn2 9011

Description: Definition of the set of positive integers. Another name for df-inn 9010. (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 9010	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 104   = wceq 1364   ∈ wcel 2167  {cab 2182  ∀wral 2475  ∩ cint 3875  (class class class)co 5925  1c1 7899   + caddc 7901  ℕcn 9009

This theorem depends on definitions:  df-inn 9010

This theorem is referenced by:  peano5nni  9012  1nn  9020  peano2nn  9021  arch  9265  caucvgre  11165