Step |
Hyp |
Ref |
Expression |
1 |
|
vex |
⊢ 𝑦 ∈ V |
2 |
1
|
elintrab |
⊢ ( 𝑦 ∈ ∩ { 𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥 } ↔ ∀ 𝑥 ∈ 𝐵 ( 𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥 ) ) |
3 |
|
ssid |
⊢ 𝐴 ⊆ 𝐴 |
4 |
|
sseq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝐴 ) ) |
5 |
|
eleq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐴 ) ) |
6 |
4 5
|
imbi12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥 ) ↔ ( 𝐴 ⊆ 𝐴 → 𝑦 ∈ 𝐴 ) ) ) |
7 |
6
|
rspcv |
⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ( 𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥 ) → ( 𝐴 ⊆ 𝐴 → 𝑦 ∈ 𝐴 ) ) ) |
8 |
3 7
|
mpii |
⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ( 𝐴 ⊆ 𝑥 → 𝑦 ∈ 𝑥 ) → 𝑦 ∈ 𝐴 ) ) |
9 |
2 8
|
syl5bi |
⊢ ( 𝐴 ∈ 𝐵 → ( 𝑦 ∈ ∩ { 𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥 } → 𝑦 ∈ 𝐴 ) ) |
10 |
9
|
ssrdv |
⊢ ( 𝐴 ∈ 𝐵 → ∩ { 𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥 } ⊆ 𝐴 ) |
11 |
|
ssintub |
⊢ 𝐴 ⊆ ∩ { 𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥 } |
12 |
11
|
a1i |
⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ⊆ ∩ { 𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥 } ) |
13 |
10 12
|
eqssd |
⊢ ( 𝐴 ∈ 𝐵 → ∩ { 𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥 } = 𝐴 ) |