Theorem dfnn2 8923

Description: Definition of the set of positive integers. Another name for df-inn 8922. (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 8922	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 104   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  ∩ cint 3846  (class class class)co 5877  1c1 7814   + caddc 7816  ℕcn 8921

This theorem depends on definitions:  df-inn 8922

This theorem is referenced by:  peano5nni  8924  1nn  8932  peano2nn  8933  arch  9175  caucvgre  10992