Theorem eltrans

Description: Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012)

	Ref	Expression
Hypothesis	eltrans.1	⊢ 𝐴 ∈ V
Assertion	eltrans	⊢ ( 𝐴 ∈ Trans ↔ Tr 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		eltrans.1	⊢ 𝐴 ∈ V
2		df-trans	⊢ Trans = ( V ∖ ran ( ( E ∘ E ) ∖ E ) )
3	2	eleq2i	⊢ ( 𝐴 ∈ Trans ↔ 𝐴 ∈ ( V ∖ ran ( ( E ∘ E ) ∖ E ) ) )
4	1	dftr6	⊢ ( Tr 𝐴 ↔ 𝐴 ∈ ( V ∖ ran ( ( E ∘ E ) ∖ E ) ) )
5	3 4	bitr4i	⊢ ( 𝐴 ∈ Trans ↔ Tr 𝐴 )