Theorem dfnn2 8041

Description: Definition of the set of positive integers. Another name for df-inn 8040. (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 8040	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 102   = wceq 1284   ∈ wcel 1433  {cab 2067  ∀wral 2348  ∩ cint 3636  (class class class)co 5532  1c1 6982   + caddc 6984  ℕcn 8039

This theorem depends on definitions:  df-inn 8040

This theorem is referenced by:  peano5nni  8042  1nn  8050  peano2nn  8051  arch  8285  caucvgre  9867