Theorem r1omhf

Description: A set is hereditarily finite iff it is finite and all of its elements are hereditarily finite. (Contributed by BTernaryTau, 19-Jan-2026)

	Ref	Expression
Assertion	r1omhf	⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ ω ) ↔ ( 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ∪ ( 𝑅1 “ ω ) ) )

Proof

Step	Hyp	Ref	Expression
1		elhf4	⊢ ( 𝐴 ∈ HF ↔ ( 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ HF ) )
2		df-hf	⊢ HF = ∪ ( 𝑅1 “ ω )
3	2	eleq2i	⊢ ( 𝐴 ∈ HF ↔ 𝐴 ∈ ∪ ( 𝑅1 “ ω ) )
4	2	eleq2i	⊢ ( 𝑥 ∈ HF ↔ 𝑥 ∈ ∪ ( 𝑅1 “ ω ) )
5	4	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 𝑥 ∈ HF ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ∪ ( 𝑅1 “ ω ) )
6	5	anbi2i	⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ HF ) ↔ ( 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ∪ ( 𝑅1 “ ω ) ) )
7	1 3 6	3bitr3i	⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ ω ) ↔ ( 𝐴 ∈ Fin ∧ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ∪ ( 𝑅1 “ ω ) ) )