Step |
Hyp |
Ref |
Expression |
1 |
|
df-fr |
⊢ ( 𝑅 Fr 𝐴 ↔ ∀ 𝑥 ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ) ) |
2 |
|
rabeq0 |
⊢ ( { 𝑧 ∈ 𝑥 ∣ 𝑧 𝑅 𝑦 } = ∅ ↔ ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ) |
3 |
2
|
rexbii |
⊢ ( ∃ 𝑦 ∈ 𝑥 { 𝑧 ∈ 𝑥 ∣ 𝑧 𝑅 𝑦 } = ∅ ↔ ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ) |
4 |
3
|
imbi2i |
⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 { 𝑧 ∈ 𝑥 ∣ 𝑧 𝑅 𝑦 } = ∅ ) ↔ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ) ) |
5 |
4
|
albii |
⊢ ( ∀ 𝑥 ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 { 𝑧 ∈ 𝑥 ∣ 𝑧 𝑅 𝑦 } = ∅ ) ↔ ∀ 𝑥 ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ∀ 𝑧 ∈ 𝑥 ¬ 𝑧 𝑅 𝑦 ) ) |
6 |
1 5
|
bitr4i |
⊢ ( 𝑅 Fr 𝐴 ↔ ∀ 𝑥 ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 { 𝑧 ∈ 𝑥 ∣ 𝑧 𝑅 𝑦 } = ∅ ) ) |