Theorem dfnn2 8722

Description: Definition of the set of positive integers. Another name for df-inn 8721. (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 8721	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 103   = wceq 1331   ∈ wcel 1480  {cab 2125  ∀wral 2416  ∩ cint 3771  (class class class)co 5774  1c1 7621   + caddc 7623  ℕcn 8720

This theorem depends on definitions:  df-inn 8721

This theorem is referenced by:  peano5nni  8723  1nn  8731  peano2nn  8732  arch  8974  caucvgre  10753