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Metamath Proof Explorer

Theorem tz6.26

Description: All nonempty subclasses of a class having a well-ordered set-like relation have minimal elements for that relation. Proposition 6.26 of TakeutiZaring p. 31. (Contributed by Scott Fenton, 29-Jan-2011) (Revised by Mario Carneiro, 26-Jun-2015) (Proof shortened by Scott Fenton, 17-Nov-2024)

Ref Expression
Assertion tz6.26 
|- ( ( ( R We A /\ R Se A ) /\ ( B C_ A /\ B =/= (/) ) ) -> E. y e. B Pred ( R , B , y ) = (/) )

Proof

Step Hyp Ref Expression
1 wefr 
 |-  ( R We A -> R Fr A )
2 1 adantr 
 |-  ( ( R We A /\ R Se A ) -> R Fr A )
3 weso 
 |-  ( R We A -> R Or A )
4 sopo 
 |-  ( R Or A -> R Po A )
5 3 4 syl 
 |-  ( R We A -> R Po A )
6 5 adantr 
 |-  ( ( R We A /\ R Se A ) -> R Po A )
7 simpr 
 |-  ( ( R We A /\ R Se A ) -> R Se A )
8 2 6 7 3jca 
 |-  ( ( R We A /\ R Se A ) -> ( R Fr A /\ R Po A /\ R Se A ) )
9 frpomin2 
 |-  ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( B C_ A /\ B =/= (/) ) ) -> E. y e. B Pred ( R , B , y ) = (/) )
10 8 9 sylan 
 |-  ( ( ( R We A /\ R Se A ) /\ ( B C_ A /\ B =/= (/) ) ) -> E. y e. B Pred ( R , B , y ) = (/) )