Theorem elhf

Description: Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015) Reduce axiom usage and shorten proof. (Revised by BJ, 27-Sep-2026)

	Ref	Expression
Assertion	elhf	⊢ ( 𝐴 ∈ HF ↔ ∃ 𝑥 ∈ ω 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) )

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	bitri	⊢ ( 𝐴 ∈ HF ↔ ∃ 𝑥 ∈ ω 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) )