Metamath Proof Explorer

Theorem sspwtrALT2

Description: Short predicate calculus proof of the right-to-left implication of dftr4 . A class which is a subclass of its power class is transitive. This proof was constructed by applying Metamath's minimize command to the proof of sspwtrALT , which is the virtual deduction proof sspwtr without virtual deductions. (Contributed by Alan Sare, 3-May-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion sspwtrALT2 A 𝒫 A Tr A


Step Hyp Ref Expression
1 ssel A 𝒫 A y A y 𝒫 A
2 1 adantld A 𝒫 A z y y A y 𝒫 A
3 elpwi y 𝒫 A y A
4 2 3 syl6 A 𝒫 A z y y A y A
5 simpl z y y A z y
6 5 a1i A 𝒫 A z y y A z y
7 ssel y A z y z A
8 4 6 7 syl6c A 𝒫 A z y y A z A
9 8 alrimivv A 𝒫 A z y z y y A z A
10 dftr2 Tr A z y z y y A z A
11 9 10 sylibr A 𝒫 A Tr A