Metamath Proof Explorer

Theorem ttcexrg

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

Ref Expression
Assertion ttcexrg 
|- ( TC+ A e. V -> A e. _V )

Proof

Step Hyp Ref Expression
1 ttcid 
 |-  A C_ TC+ A
2 ssexg 
 |-  ( ( A C_ TC+ A /\ TC+ A e. V ) -> A e. _V )
3 1 2 mpan 
 |-  ( TC+ A e. V -> A e. _V )