Theorem r1omfi

Description: Obsolete theorem, use hffi instead. Hereditarily finite sets are finite sets. (Contributed by BTernaryTau, 30-Dec-2025) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	r1omfi	⊢ ∪ ( 𝑅1 “ ω ) ⊆ Fin

Proof

Step	Hyp	Ref	Expression
1		df-hf	⊢ HF = ∪ ( 𝑅1 “ ω )
2	1	eleq2i	⊢ ( 𝑥 ∈ HF ↔ 𝑥 ∈ ∪ ( 𝑅1 “ ω ) )
3		hffi	⊢ ( 𝑥 ∈ HF → 𝑥 ∈ Fin )
4	2 3	sylbir	⊢ ( 𝑥 ∈ ∪ ( 𝑅1 “ ω ) → 𝑥 ∈ Fin )
5	4	ssriv	⊢ ∪ ( 𝑅1 “ ω ) ⊆ Fin