Theorem dfnn2 9123

Description: Definition of the set of positive integers. Another name for df-inn 9122. (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 9122	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 104   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∩ cint 3923  (class class class)co 6007  1c1 8011   + caddc 8013  ℕcn 9121

This theorem depends on definitions:  df-inn 9122

This theorem is referenced by:  peano5nni  9124  1nn  9132  peano2nn  9133  arch  9377  caucvgre  11507