Theorem dfnn2 8880

Description: Definition of the set of positive integers. Another name for df-inn 8879. (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 8879	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 103   = wceq 1348   ∈ wcel 2141  {cab 2156  ∀wral 2448  ∩ cint 3831  (class class class)co 5853  1c1 7775   + caddc 7777  ℕcn 8878

This theorem depends on definitions:  df-inn 8879

This theorem is referenced by:  peano5nni  8881  1nn  8889  peano2nn  8890  arch  9132  caucvgre  10945