Theorem dftr4

Description: An alternate way of defining a transitive class. Definition of Enderton p. 71. (Contributed by NM, 29-Aug-1993)

	Ref	Expression
Assertion	dftr4	⊢ ( Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		df-tr	⊢ ( Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴 )
2		sspwuni	⊢ ( 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴 )
3	1 2	bitr4i	⊢ ( Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴 )