Metamath Proof Explorer

Theorem hfsn

Description: The singleton of a hereditarily finite set is a hereditarily finite set. (Contributed by Scott Fenton, 15-Jul-2015) Avoid ax-reg , ax-inf2 . (Revised by BTernaryTau, 17-Sep-2026)

Ref Expression
Assertion hfsn ⊢ A ∈ Hf → A ∈ Hf

Proof

Step Hyp Ref Expression
1 snfi ⊢ A ∈ Fin
2 snssi ⊢ A ∈ Hf → A ⊆ Hf
3 elhf3 ⊢ A ∈ Hf ↔ A ∈ Fin ∧ A ⊆ Hf
4 3 biimpri ⊢ A ∈ Fin ∧ A ⊆ Hf → A ∈ Hf
5 1 2 4 sylancr ⊢ A ∈ Hf → A ∈ Hf