Step |
Hyp |
Ref |
Expression |
1 |
|
onelon |
⊢ ( ( 𝐵 ∈ On ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ On ) |
2 |
|
onelon |
⊢ ( ( 𝐵 ∈ On ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ On ) |
3 |
1 2
|
anim12dan |
⊢ ( ( 𝐵 ∈ On ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ∈ On ∧ 𝑦 ∈ On ) ) |
4 |
3
|
ex |
⊢ ( 𝐵 ∈ On → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ∈ On ∧ 𝑦 ∈ On ) ) ) |
5 |
|
onin |
⊢ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ On ) → ( 𝑥 ∩ 𝑦 ) ∈ On ) |
6 |
4 5
|
syl6 |
⊢ ( 𝐵 ∈ On → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ∩ 𝑦 ) ∈ On ) ) |
7 |
6
|
anc2ri |
⊢ ( 𝐵 ∈ On → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 ∩ 𝑦 ) ∈ On ∧ 𝐵 ∈ On ) ) ) |
8 |
|
inss1 |
⊢ ( 𝑥 ∩ 𝑦 ) ⊆ 𝑥 |
9 |
8
|
jctl |
⊢ ( 𝑥 ∈ 𝐵 → ( ( 𝑥 ∩ 𝑦 ) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ) |
10 |
9
|
adantr |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 ∩ 𝑦 ) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ) |
11 |
10
|
a1i |
⊢ ( 𝐵 ∈ On → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 ∩ 𝑦 ) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵 ) ) ) |
12 |
|
ontr2 |
⊢ ( ( ( 𝑥 ∩ 𝑦 ) ∈ On ∧ 𝐵 ∈ On ) → ( ( ( 𝑥 ∩ 𝑦 ) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 ∩ 𝑦 ) ∈ 𝐵 ) ) |
13 |
7 11 12
|
syl6c |
⊢ ( 𝐵 ∈ On → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ∩ 𝑦 ) ∈ 𝐵 ) ) |
14 |
13
|
ralrimivv |
⊢ ( 𝐵 ∈ On → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ∩ 𝑦 ) ∈ 𝐵 ) |
15 |
|
fiinbas |
⊢ ( ( 𝐵 ∈ On ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ∩ 𝑦 ) ∈ 𝐵 ) → 𝐵 ∈ TopBases ) |
16 |
14 15
|
mpdan |
⊢ ( 𝐵 ∈ On → 𝐵 ∈ TopBases ) |