Theorem dfnn2 9309

Description: Definition of the set of positive integers. Another name for df-inn 9308. (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 9308	1 ⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

This proof depends on syntax axioms:   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∩ cint 3970  (class class class)co 6085  1c1 8181   + caddc 8183  ℕcn 9307

This proof depends on definitions:  df-inn 9308

This theorem is used by:  peano5nni  9310  1nn  9318  peano2nn  9319  arch  9565  caucvgre  11763