Theorem ttcpwss

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

	Ref	Expression
Assertion	ttcpwss	|- TC+ ~P A C_ ~P TC+ A

Proof

Step	Hyp	Ref	Expression
1		ttcid	|- A C_ TC+ A
2	1	sspwi	|- ~P A C_ ~P TC+ A
3		ttctr	|- Tr TC+ A
4		pwtr	|- ( Tr TC+ A <-> Tr ~P TC+ A )
5	3 4	mpbi	|- Tr ~P TC+ A
6		ttcmin	|- ( ( ~P A C_ ~P TC+ A /\ Tr ~P TC+ A ) -> TC+ ~P A C_ ~P TC+ A )
7	2 5 6	mp2an	|- TC+ ~P A C_ ~P TC+ A