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	Could not format assertion : No typesetting found for |- ( A e. HF -> ~P A e. HF ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		hffi	Could not format ( A e. HF -> A e. Fin ) : No typesetting found for |- ( A e. HF -> A e. Fin ) with typecode |-
2		pwfi	$$⊢A\in \mathrm{Fin}\leftrightarrow 𝒫A\in \mathrm{Fin}$$
3	1 2	sylib	Could not format ( A e. HF -> ~P A e. Fin ) : No typesetting found for |- ( A e. HF -> ~P A e. Fin ) with typecode |-
4		hfsshf	Could not format ( ( x C_ A /\ A e. HF ) -> x e. HF ) : No typesetting found for |- ( ( x C_ A /\ A e. HF ) -> x e. HF ) with typecode |-
5	4	expcom	Could not format ( A e. HF -> ( x C_ A -> x e. HF ) ) : No typesetting found for |- ( A e. HF -> ( x C_ A -> x e. HF ) ) with typecode |-
6	5	alrimiv	Could not format ( A e. HF -> A. x ( x C_ A -> x e. HF ) ) : No typesetting found for |- ( A e. HF -> A. x ( x C_ A -> x e. HF ) ) with typecode |-
7		pwss	Could not format ( ~P A C_ HF <-> A. x ( x C_ A -> x e. HF ) ) : No typesetting found for |- ( ~P A C_ HF <-> A. x ( x C_ A -> x e. HF ) ) with typecode |-
8	6 7	sylibr	Could not format ( A e. HF -> ~P A C_ HF ) : No typesetting found for |- ( A e. HF -> ~P A C_ HF ) with typecode |-
9		elhf3	Could not format ( ~P A e. HF <-> ( ~P A e. Fin /\ ~P A C_ HF ) ) : No typesetting found for |- ( ~P A e. HF <-> ( ~P A e. Fin /\ ~P A C_ HF ) ) with typecode |-
10	3 8 9	sylanbrc	Could not format ( A e. HF -> ~P A e. HF ) : No typesetting found for |- ( A e. HF -> ~P A e. HF ) with typecode |-