Theorem dfnn2 8859

Description: Definition of the set of positive integers. Another name for df-inn 8858. (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 8858	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 103   = wceq 1343   ∈ wcel 2136  {cab 2151  ∀wral 2444  ∩ cint 3824  (class class class)co 5842  1c1 7754   + caddc 7756  ℕcn 8857

This theorem depends on definitions:  df-inn 8858

This theorem is referenced by:  peano5nni  8860  1nn  8868  peano2nn  8869  arch  9111  caucvgre  10923