Theorem dfnn2 9058

Description: Definition of the set of positive integers. Another name for df-inn 9057. (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 9057	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 104   = wceq 1373   ∈ wcel 2177  {cab 2192  ∀wral 2485  ∩ cint 3891  (class class class)co 5957  1c1 7946   + caddc 7948  ℕcn 9056

This theorem depends on definitions:  df-inn 9057

This theorem is referenced by:  peano5nni  9059  1nn  9067  peano2nn  9068  arch  9312  caucvgre  11367