Theorem dfnn2 9108

Description: Definition of the set of positive integers. Another name for df-inn 9107. (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 9107	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 104   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∩ cint 3922  (class class class)co 6000  1c1 7996   + caddc 7998  ℕcn 9106

This theorem depends on definitions:  df-inn 9107

This theorem is referenced by:  peano5nni  9109  1nn  9117  peano2nn  9118  arch  9362  caucvgre  11487