Theorem hfadj

Description: Adjoining one hereditarily finite set to a hereditarily finite set results in a hereditarily finite set. (Contributed by Scott Fenton, 15-Jul-2015)

	Ref	Expression
Assertion	hfadj	|- ( ( A e. HF /\ B e. HF ) -> ( A u. { B } ) e. HF )

Proof

Step	Hyp	Ref	Expression
1		hfsn	|- ( B e. HF -> { B } e. HF )
2		hfun	|- ( ( A e. HF /\ { B } e. HF ) -> ( A u. { B } ) e. HF )
3	1 2	sylan2	|- ( ( A e. HF /\ B e. HF ) -> ( A u. { B } ) e. HF )