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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 C_ A /\ B =/= (/) ) -> E. y e. 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 C_ A /\ B =/= (/) ) ) -> E. y e. B Pred ( R , B , y ) = (/) )
4	1 2 3	mpanl12	\|- ( ( B C_ A /\ B =/= (/) ) -> E. y e. B Pred ( R , B , y ) = (/) )