Theorem hfpw

Description: The power class of a hereditarily finite set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015) Avoid ax-reg , ax-inf2 . (Revised by BTernaryTau, 17-Sep-2026)

	Ref	Expression
Assertion	hfpw	|- ( A e. HF -> ~P A e. HF )

Proof

Step	Hyp	Ref	Expression
1		hffi	|- ( A e. HF -> A e. Fin )
2		pwfi	|- ( A e. Fin <-> ~P A e. Fin )
3	1 2	sylib	|- ( A e. HF -> ~P A e. Fin )
4		hfsshf	|- ( ( x C_ A /\ A e. HF ) -> x e. HF )
5	4	expcom	|- ( A e. HF -> ( x C_ A -> x e. HF ) )
6	5	alrimiv	|- ( A e. HF -> A. x ( x C_ A -> x e. HF ) )
7		pwss	|- ( ~P A C_ HF <-> A. x ( x C_ A -> x e. HF ) )
8	6 7	sylibr	|- ( A e. HF -> ~P A C_ HF )
9		elhf3	|- ( ~P A e. HF <-> ( ~P A e. Fin /\ ~P A C_ HF ) )
10	3 8 9	sylanbrc	|- ( A e. HF -> ~P A e. HF )