Metamath Proof Explorer


Theorem tz6.26i

Description: All nonempty subclasses of a class having a well-ordered set-like relation R have R-minimal elements. Proposition 6.26 of TakeutiZaring p. 31. (Contributed by Scott Fenton, 14-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015)

Ref Expression
Hypotheses tz6.26i.1 R We A
tz6.26i.2 R Se A
Assertion tz6.26i B A B y B Pred R B y =

Proof

Step Hyp Ref Expression
1 tz6.26i.1 R We A
2 tz6.26i.2 R Se A
3 tz6.26 R We A R Se A B A B y B Pred R B y =
4 1 2 3 mpanl12 B A B y B Pred R B y =