Theorem dfnn2 9144

Description: Definition of the set of positive integers. Another name for df-inn 9143. (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 9143	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 104   = wceq 1397   ∈ wcel 2202  {cab 2217  ∀wral 2510  ∩ cint 3928  (class class class)co 6017  1c1 8032   + caddc 8034  ℕcn 9142

This theorem depends on definitions:  df-inn 9143

This theorem is referenced by:  peano5nni  9145  1nn  9153  peano2nn  9154  arch  9398  caucvgre  11541