Theorem dfnn2 9285

Description: Definition of the set of positive integers. Another name for df-inn 9284. (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 9284	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∩ cint 3965  (class class class)co 6075  1c1 8170   + caddc 8172  ℕcn 9283

This theorem depends on definitions:  df-inn 9284

This theorem is referenced by:  peano5nni  9286  1nn  9294  peano2nn  9295  arch  9539  caucvgre  11725