Theorem succ_in_onat

Proof
let inductive A ↔ ∅ ∈ A ∧ (∀n. n ∈ A → n ∪ {n} ∈ A).
01. 1 ⊢ n ∈ onat, hypo.
02. 2 ⊢ A ∈ {A | inductive A}, hypo.
03. 2 ⊢ ∅ ∈ A ∧ ∀n. n ∈ A → n ∪ {n} ∈ A, comp_elim 2.
04. 2 ⊢ ∀n. n ∈ A → n ∪ {n} ∈ A, conj_elimr 3.
05. 2 ⊢ onat ⊆ A, Intersection_is_lower_bound onat_eq 2.
06. 1, 2 ⊢ n ∈ A, incl_elim 5 1.
07. 1, 2 ⊢ n ∪ {n} ∈ A, uq_bounded_elim 4 6.
08. 1, 2 ⊢ succ n ∈ A, eq_subst_rev succ_eq 7, P x ↔ x ∈ A.
09. 1 ⊢ A ∈ {A | inductive A} → succ n ∈ A, subj_intro 8.
10. 1 ⊢ ∀A. A ∈ {A | inductive A} → succ n ∈ A, uq_intro 9.
11. 1 ⊢ set n, set_intro 1.
12. 1 ⊢ set (succ n), succ_is_set 11.
13. 1 ⊢ succ n ∈ ⋂{A | inductive A}, Intersection_intro 12 10.
14. 1 ⊢ succ n ∈ onat, eq_subst_rev onat_eq 13, P x ↔ succ n ∈ x.
succ_in_onat. ⊢ n ∈ onat → succ n ∈ onat, subj_intro 14.

Dependencies
The given proof depends on six axioms:
comp, eq_refl, eq_subst, ext, pairing, union.