Step |
Hyp |
Ref |
Expression |
1 |
|
dfcleq |
⊢ ( { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) } = 𝐴 ↔ ∀ 𝑥 ( 𝑥 ∈ { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) } ↔ 𝑥 ∈ 𝐴 ) ) |
2 |
|
df-in |
⊢ ( 𝐴 ∩ 𝐵 ) = { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) } |
3 |
2
|
eqeq1i |
⊢ ( ( 𝐴 ∩ 𝐵 ) = 𝐴 ↔ { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) } = 𝐴 ) |
4 |
|
df-ss |
⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ) |
5 |
|
simp2 |
⊢ ( ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐴 ) |
6 |
5
|
3expib |
⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐴 ) ) |
7 |
|
ancl |
⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ) ) |
8 |
6 7
|
impbid |
⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ↔ 𝑥 ∈ 𝐴 ) ) |
9 |
|
dfbi2 |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ↔ 𝑥 ∈ 𝐴 ) ↔ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐴 ) ∧ ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ) ) ) |
10 |
|
pm2.21 |
⊢ ( ¬ 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ) |
11 |
|
pm3.4 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ) |
12 |
10 11
|
ja |
⊢ ( ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ) |
13 |
9 12
|
simplbiim |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ↔ 𝑥 ∈ 𝐴 ) → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ) |
14 |
8 13
|
impbii |
⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ↔ 𝑥 ∈ 𝐴 ) ) |
15 |
|
df-clab |
⊢ ( 𝑥 ∈ { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) } ↔ [ 𝑥 / 𝑦 ] ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) |
16 |
|
eleq1w |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴 ) ) |
17 |
|
eleq1w |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵 ) ) |
18 |
16 17
|
anbi12d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ) ) |
19 |
18
|
sbievw |
⊢ ( [ 𝑥 / 𝑦 ] ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ) |
20 |
15 19
|
bitr2i |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ↔ 𝑥 ∈ { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) } ) |
21 |
20
|
bibi1i |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) ↔ 𝑥 ∈ 𝐴 ) ↔ ( 𝑥 ∈ { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) } ↔ 𝑥 ∈ 𝐴 ) ) |
22 |
14 21
|
bitri |
⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ↔ ( 𝑥 ∈ { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) } ↔ 𝑥 ∈ 𝐴 ) ) |
23 |
22
|
albii |
⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ↔ ∀ 𝑥 ( 𝑥 ∈ { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) } ↔ 𝑥 ∈ 𝐴 ) ) |
24 |
4 23
|
bitri |
⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) } ↔ 𝑥 ∈ 𝐴 ) ) |
25 |
1 3 24
|
3bitr4ri |
⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) = 𝐴 ) |