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	$$⊢A\in \bigcup R_1\left[\mathrm{\omega }\right]\leftrightarrow \left(A\in \mathrm{Fin}\wedge \forall x\in Ax\in \bigcup R_1\left[\mathrm{\omega }\right]\right)$$

Proof

Step	Hyp	Ref	Expression
1		elhf4	Could not format ( A e. HF <-> ( A e. Fin /\ A. x e. A x e. HF ) ) : No typesetting found for |- ( A e. HF <-> ( A e. Fin /\ A. x e. A x e. HF ) ) with typecode |-
2		df-hf	Could not format HF = U. ( R1 " _om ) : No typesetting found for |- HF = U. ( R1 " _om ) with typecode |-
3	2	eleq2i	Could not format ( A e. HF <-> A e. U. ( R1 " _om ) ) : No typesetting found for |- ( A e. HF <-> A e. U. ( R1 " _om ) ) with typecode |-
4	2	eleq2i	Could not format ( x e. HF <-> x e. U. ( R1 " _om ) ) : No typesetting found for |- ( x e. HF <-> x e. U. ( R1 " _om ) ) with typecode |-
5	4	ralbii	Could not format ( A. x e. A x e. HF <-> A. x e. A x e. U. ( R1 " _om ) ) : No typesetting found for |- ( A. x e. A x e. HF <-> A. x e. A x e. U. ( R1 " _om ) ) with typecode |-
6	5	anbi2i	Could not format ( ( A e. Fin /\ A. x e. A x e. HF ) <-> ( A e. Fin /\ A. x e. A x e. U. ( R1 " _om ) ) ) : No typesetting found for |- ( ( A e. Fin /\ A. x e. A x e. HF ) <-> ( A e. Fin /\ A. x e. A x e. U. ( R1 " _om ) ) ) with typecode |-
7	1 3 6	3bitr3i	$$⊢A\in \bigcup R_1\left[\mathrm{\omega }\right]\leftrightarrow \left(A\in \mathrm{Fin}\wedge \forall x\in Ax\in \bigcup R_1\left[\mathrm{\omega }\right]\right)$$