Metamath Proof Explorer


Theorem fri

Description: A nonempty subset of an R -well-founded class has an R -minimal element (inference form). (Contributed by BJ, 16-Nov-2024) (Proof shortened by BJ, 19-Nov-2024)

Ref Expression
Assertion fri ( ( ( 𝐵𝐶𝑅 Fr 𝐴 ) ∧ ( 𝐵𝐴𝐵 ≠ ∅ ) ) → ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )

Proof

Step Hyp Ref Expression
1 simplr ( ( ( 𝐵𝐶𝑅 Fr 𝐴 ) ∧ ( 𝐵𝐴𝐵 ≠ ∅ ) ) → 𝑅 Fr 𝐴 )
2 simprl ( ( ( 𝐵𝐶𝑅 Fr 𝐴 ) ∧ ( 𝐵𝐴𝐵 ≠ ∅ ) ) → 𝐵𝐴 )
3 simpll ( ( ( 𝐵𝐶𝑅 Fr 𝐴 ) ∧ ( 𝐵𝐴𝐵 ≠ ∅ ) ) → 𝐵𝐶 )
4 simprr ( ( ( 𝐵𝐶𝑅 Fr 𝐴 ) ∧ ( 𝐵𝐴𝐵 ≠ ∅ ) ) → 𝐵 ≠ ∅ )
5 1 2 3 4 frd ( ( ( 𝐵𝐶𝑅 Fr 𝐴 ) ∧ ( 𝐵𝐴𝐵 ≠ ∅ ) ) → ∃ 𝑥𝐵𝑦𝐵 ¬ 𝑦 𝑅 𝑥 )