Theorem hffi

Description: Hereditarily finite sets are finite sets. (Contributed by BTernaryTau, 30-Dec-2025) Restate using the defined HF symbol. (Revised by Eric Schmidt, 8-Sep-2026)

	Ref	Expression
Assertion	hffi	⊢ ( 𝐴 ∈ HF → 𝐴 ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		df-hf	⊢ HF = ∪ ( 𝑅1 “ ω )
2	1	eleq2i	⊢ ( 𝐴 ∈ HF ↔ 𝐴 ∈ ∪ ( 𝑅1 “ ω ) )
3		r1fun	⊢ Fun 𝑅1
4		eluniima	⊢ ( Fun 𝑅1 → ( 𝐴 ∈ ∪ ( 𝑅1 “ ω ) ↔ ∃ 𝑥 ∈ ω 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) ) )
5	3 4	ax-mp	⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ ω ) ↔ ∃ 𝑥 ∈ ω 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) )
6	2 5	sylbb	⊢ ( 𝐴 ∈ HF → ∃ 𝑥 ∈ ω 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) )
7		r1fin	⊢ ( 𝑥 ∈ ω → ( 𝑅1 ‘ 𝑥 ) ∈ Fin )
8		r1pwss	⊢ ( 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) → 𝒫 𝐴 ⊆ ( 𝑅1 ‘ 𝑥 ) )
9		ssfi	⊢ ( ( ( 𝑅1 ‘ 𝑥 ) ∈ Fin ∧ 𝒫 𝐴 ⊆ ( 𝑅1 ‘ 𝑥 ) ) → 𝒫 𝐴 ∈ Fin )
10	7 8 9	syl2an	⊢ ( ( 𝑥 ∈ ω ∧ 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) ) → 𝒫 𝐴 ∈ Fin )
11	10	rexlimiva	⊢ ( ∃ 𝑥 ∈ ω 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) → 𝒫 𝐴 ∈ Fin )
12		pwfir	⊢ ( 𝒫 𝐴 ∈ Fin → 𝐴 ∈ Fin )
13	6 11 12	3syl	⊢ ( 𝐴 ∈ HF → 𝐴 ∈ Fin )