Theorem pwtr

Description: A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011) (Revised by Mario Carneiro, 15-Jun-2014)

	Ref	Expression
Assertion	pwtr	⊢ ( Tr 𝐴 ↔ Tr 𝒫 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		unipw	⊢ ∪ 𝒫 𝐴 = 𝐴
2	1	sseq1i	⊢ ( ∪ 𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴 )
3		df-tr	⊢ ( Tr 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝒫 𝐴 )
4		dftr4	⊢ ( Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴 )
5	2 3 4	3bitr4ri	⊢ ( Tr 𝐴 ↔ Tr 𝒫 𝐴 )