Metamath Proof Explorer


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 𝐴𝐴 ∈ ( V ∖ ran ( ( E ∘ E ) ∖ E ) ) )
5 3 4 bitr4i ( 𝐴 Trans ↔ Tr 𝐴 )