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	⊢ ( ( ( 𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		simplr	⊢ ( ( ( 𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → 𝑅 Fr 𝐴 )
2		simprl	⊢ ( ( ( 𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → 𝐵 ⊆ 𝐴 )
3		simpll	⊢ ( ( ( 𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → 𝐵 ∈ 𝐶 )
4		simprr	⊢ ( ( ( 𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → 𝐵 ≠ ∅ )
5	1 2 3 4	frd	⊢ ( ( ( 𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 )