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 A A V ran E E E
5 3 4 bitr4i A 𝖳𝗋𝖺𝗇𝗌 Tr A