Theorem hfpw

Description: The power class of a hereditarily finite set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015) Avoid ax-reg , ax-inf2 . (Revised by BTernaryTau, 17-Sep-2026)

	Ref	Expression
Assertion	hfpw	⊢ ( 𝐴 ∈ HF → 𝒫 𝐴 ∈ HF )

Proof

Step	Hyp	Ref	Expression
1		hffi	⊢ ( 𝐴 ∈ HF → 𝐴 ∈ Fin )
2		pwfi	⊢ ( 𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin )
3	1 2	sylib	⊢ ( 𝐴 ∈ HF → 𝒫 𝐴 ∈ Fin )
4		hfsshf	⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝐴 ∈ HF ) → 𝑥 ∈ HF )
5	4	expcom	⊢ ( 𝐴 ∈ HF → ( 𝑥 ⊆ 𝐴 → 𝑥 ∈ HF ) )
6	5	alrimiv	⊢ ( 𝐴 ∈ HF → ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → 𝑥 ∈ HF ) )
7		pwss	⊢ ( 𝒫 𝐴 ⊆ HF ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → 𝑥 ∈ HF ) )
8	6 7	sylibr	⊢ ( 𝐴 ∈ HF → 𝒫 𝐴 ⊆ HF )
9		elhf3	⊢ ( 𝒫 𝐴 ∈ HF ↔ ( 𝒫 𝐴 ∈ Fin ∧ 𝒫 𝐴 ⊆ HF ) )
10	3 8 9	sylanbrc	⊢ ( 𝐴 ∈ HF → 𝒫 𝐴 ∈ HF )