Theorem ttcsnexg

Description: If the transitive closure of a class is a set, then its singleton transitive closure is a set. (Contributed by Matthew House, 6-Apr-2026)

	Ref	Expression
Assertion	ttcsnexg	|- ( TC+ A e. V -> TC+ { A } e. _V )

Proof

Step	Hyp	Ref	Expression
1		ttcexrg	|- ( TC+ A e. V -> A e. _V )
2		ttcsng	|- ( A e. _V -> TC+ { A } = ( TC+ A u. { A } ) )
3	1 2	syl	|- ( TC+ A e. V -> TC+ { A } = ( TC+ A u. { A } ) )
4		snex	|- { A } e. _V
5		unexg	|- ( ( TC+ A e. V /\ { A } e. _V ) -> ( TC+ A u. { A } ) e. _V )
6	4 5	mpan2	|- ( TC+ A e. V -> ( TC+ A u. { A } ) e. _V )
7	3 6	eqeltrd	|- ( TC+ A e. V -> TC+ { A } e. _V )