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 ) |
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 ) |