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 ( ( 𝐵𝐴𝐵 ≠ ∅ ) → ∃ 𝑦𝐵 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 𝐴𝑅 Se 𝐴 ) ∧ ( 𝐵𝐴𝐵 ≠ ∅ ) ) → ∃ 𝑦𝐵 Pred ( 𝑅 , 𝐵 , 𝑦 ) = ∅ )
4 1 2 3 mpanl12 ( ( 𝐵𝐴𝐵 ≠ ∅ ) → ∃ 𝑦𝐵 Pred ( 𝑅 , 𝐵 , 𝑦 ) = ∅ )