Theorem dfnn2 8317

Description: Definition of the set of positive integers. Another name for df-inn 8316. (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 8316	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Syntax hints:   ∧ wa 102   = wceq 1285   ∈ wcel 1434  {cab 2069  ∀wral 2353  ∩ cint 3662  (class class class)co 5590  1c1 7253   + caddc 7255  ℕcn 8315

This theorem depends on definitions:  df-inn 8316

This theorem is referenced by:  peano5nni  8318  1nn  8326  peano2nn  8327  arch  8561  caucvgre  10240