Step |
Hyp |
Ref |
Expression |
1 |
|
dffr2 |
⊢ ( E Fr 𝐴 ↔ ∀ 𝑥 ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 { 𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦 } = ∅ ) ) |
2 |
|
epel |
⊢ ( 𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦 ) |
3 |
2
|
rabbii |
⊢ { 𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦 } = { 𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦 } |
4 |
|
dfin5 |
⊢ ( 𝑥 ∩ 𝑦 ) = { 𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦 } |
5 |
3 4
|
eqtr4i |
⊢ { 𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦 } = ( 𝑥 ∩ 𝑦 ) |
6 |
5
|
eqeq1i |
⊢ ( { 𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦 } = ∅ ↔ ( 𝑥 ∩ 𝑦 ) = ∅ ) |
7 |
6
|
rexbii |
⊢ ( ∃ 𝑦 ∈ 𝑥 { 𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦 } = ∅ ↔ ∃ 𝑦 ∈ 𝑥 ( 𝑥 ∩ 𝑦 ) = ∅ ) |
8 |
7
|
imbi2i |
⊢ ( ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 { 𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦 } = ∅ ) ↔ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ( 𝑥 ∩ 𝑦 ) = ∅ ) ) |
9 |
8
|
albii |
⊢ ( ∀ 𝑥 ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 { 𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦 } = ∅ ) ↔ ∀ 𝑥 ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ( 𝑥 ∩ 𝑦 ) = ∅ ) ) |
10 |
1 9
|
bitri |
⊢ ( E Fr 𝐴 ↔ ∀ 𝑥 ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑥 ( 𝑥 ∩ 𝑦 ) = ∅ ) ) |