Theorem ttcel

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

	Ref	Expression
Assertion	ttcel	⊢ ( 𝐴 ∈ TC+ 𝐵 → TC+ 𝐴 ⊆ TC+ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ttctr2	⊢ ( 𝐴 ∈ TC+ 𝐵 → 𝐴 ⊆ TC+ 𝐵 )
2		ttctr	⊢ Tr TC+ 𝐵
3		ttcmin	⊢ ( ( 𝐴 ⊆ TC+ 𝐵 ∧ Tr TC+ 𝐵 ) → TC+ 𝐴 ⊆ TC+ 𝐵 )
4	1 2 3	sylancl	⊢ ( 𝐴 ∈ TC+ 𝐵 → TC+ 𝐴 ⊆ TC+ 𝐵 )