Metamath Proof Explorer

Theorem ttcss

Description: A transitive closure contains the transitive closures of all its subclasses. (Contributed by Matthew House, 6-Apr-2026)

Ref Expression
Assertion ttcss 
|- ( A C_ TC+ B -> TC+ A C_ TC+ B )

Proof

Step Hyp Ref Expression
1 ttctr 
 |-  Tr TC+ B
2 ttcmin 
 |-  ( ( A C_ TC+ B /\ Tr TC+ B ) -> TC+ A C_ TC+ B )
3 1 2 mpan2 
 |-  ( A C_ TC+ B -> TC+ A C_ TC+ B )