Theorem dfnn2 7916

Description: Definition of the set of positive integers. Another name for df-inn 7915. (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 7915	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 97   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∩ cint 3615  (class class class)co 5512  1c1 6890   + caddc 6892  ℕcn 7914

This theorem depends on definitions:  df-inn 7915

This theorem is referenced by:  peano5nni  7917  1nn  7925  peano2nn  7926  arch  8178  caucvgre  9580