Theorem dfnn2 8850

Description: Definition of the set of positive integers. Another name for df-inn 8849. (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 8849	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 103   = wceq 1342   ∈ wcel 2135  {cab 2150  ∀wral 2442  ∩ cint 3818  (class class class)co 5836  1c1 7745   + caddc 7747  ℕcn 8848

This theorem depends on definitions:  df-inn 8849

This theorem is referenced by:  peano5nni  8851  1nn  8859  peano2nn  8860  arch  9102  caucvgre  10909