Step |
Hyp |
Ref |
Expression |
1 |
|
elsni |
⊢ ( 𝑥 ∈ { 𝐴 } → 𝑥 = 𝐴 ) |
2 |
|
elsni |
⊢ ( 𝑦 ∈ { 𝐴 } → 𝑦 = 𝐴 ) |
3 |
1 2
|
ineqan12d |
⊢ ( ( 𝑥 ∈ { 𝐴 } ∧ 𝑦 ∈ { 𝐴 } ) → ( 𝑥 ∩ 𝑦 ) = ( 𝐴 ∩ 𝐴 ) ) |
4 |
|
inidm |
⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴 |
5 |
3 4
|
eqtrdi |
⊢ ( ( 𝑥 ∈ { 𝐴 } ∧ 𝑦 ∈ { 𝐴 } ) → ( 𝑥 ∩ 𝑦 ) = 𝐴 ) |
6 |
|
vex |
⊢ 𝑥 ∈ V |
7 |
6
|
inex1 |
⊢ ( 𝑥 ∩ 𝑦 ) ∈ V |
8 |
7
|
elsn |
⊢ ( ( 𝑥 ∩ 𝑦 ) ∈ { 𝐴 } ↔ ( 𝑥 ∩ 𝑦 ) = 𝐴 ) |
9 |
5 8
|
sylibr |
⊢ ( ( 𝑥 ∈ { 𝐴 } ∧ 𝑦 ∈ { 𝐴 } ) → ( 𝑥 ∩ 𝑦 ) ∈ { 𝐴 } ) |
10 |
9
|
rgen2 |
⊢ ∀ 𝑥 ∈ { 𝐴 } ∀ 𝑦 ∈ { 𝐴 } ( 𝑥 ∩ 𝑦 ) ∈ { 𝐴 } |
11 |
|
snex |
⊢ { 𝐴 } ∈ V |
12 |
|
inficl |
⊢ ( { 𝐴 } ∈ V → ( ∀ 𝑥 ∈ { 𝐴 } ∀ 𝑦 ∈ { 𝐴 } ( 𝑥 ∩ 𝑦 ) ∈ { 𝐴 } ↔ ( fi ‘ { 𝐴 } ) = { 𝐴 } ) ) |
13 |
11 12
|
ax-mp |
⊢ ( ∀ 𝑥 ∈ { 𝐴 } ∀ 𝑦 ∈ { 𝐴 } ( 𝑥 ∩ 𝑦 ) ∈ { 𝐴 } ↔ ( fi ‘ { 𝐴 } ) = { 𝐴 } ) |
14 |
10 13
|
mpbi |
⊢ ( fi ‘ { 𝐴 } ) = { 𝐴 } |