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Definition df-inn 8733
Description: Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 8734 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)
Assertion
Ref Expression
df-inn ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-inn
StepHypRef Expression
1 cn 8732 . 2 class
2 c1 7633 . . . . . 6 class 1
3 vx . . . . . . 7 setvar 𝑥
43cv 1330 . . . . . 6 class 𝑥
52, 4wcel 1480 . . . . 5 wff 1 ∈ 𝑥
6 vy . . . . . . . . 9 setvar 𝑦
76cv 1330 . . . . . . . 8 class 𝑦
8 caddc 7635 . . . . . . . 8 class +
97, 2, 8co 5774 . . . . . . 7 class (𝑦 + 1)
109, 4wcel 1480 . . . . . 6 wff (𝑦 + 1) ∈ 𝑥
1110, 6, 4wral 2416 . . . . 5 wff 𝑦𝑥 (𝑦 + 1) ∈ 𝑥
125, 11wa 103 . . . 4 wff (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)
1312, 3cab 2125 . . 3 class {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
1413cint 3771 . 2 class {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
151, 14wceq 1331 1 wff ℕ = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Colors of variables: wff set class
This definition is referenced by:  dfnn2  8734
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