Theorem dfnn2 8522

Description: Definition of the set of positive integers. Another name for df-inn 8521. (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 8521	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 103   = wceq 1296   ∈ wcel 1445  {cab 2081  ∀wral 2370  ∩ cint 3710  (class class class)co 5690  1c1 7448   + caddc 7450  ℕcn 8520

This theorem depends on definitions:  df-inn 8521

This theorem is referenced by:  peano5nni  8523  1nn  8531  peano2nn  8532  arch  8768  caucvgre  10529