Theorem dfnn2 8986

Description: Definition of the set of positive integers. Another name for df-inn 8985. (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 8985	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 104   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∩ cint 3871  (class class class)co 5919  1c1 7875   + caddc 7877  ℕcn 8984

This theorem depends on definitions:  df-inn 8985

This theorem is referenced by:  peano5nni  8987  1nn  8995  peano2nn  8996  arch  9240  caucvgre  11128