Metamath Proof Explorer

Theorem ttctrid

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

Ref Expression
Assertion ttctrid 
|- ( Tr A -> TC+ A = A )

Proof

Step Hyp Ref Expression
1 ssid 
 |-  A C_ A
2 ttcmin 
 |-  ( ( A C_ A /\ Tr A ) -> TC+ A C_ A )
3 1 2 mpan 
 |-  ( Tr A -> TC+ A C_ A )
4 ttcid 
 |-  A C_ TC+ A
5 4 a1i 
 |-  ( Tr A -> A C_ TC+ A )
6 3 5 eqssd 
 |-  ( Tr A -> TC+ A = A )