Step |
Hyp |
Ref |
Expression |
1 |
|
uniexg |
⊢ ( 𝐴 ∈ V → ∪ 𝐴 ∈ V ) |
2 |
1
|
pwexd |
⊢ ( 𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ V ) |
3 |
|
pwuni |
⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 |
4 |
|
fiss |
⊢ ( ( 𝒫 ∪ 𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴 ) → ( fi ‘ 𝐴 ) ⊆ ( fi ‘ 𝒫 ∪ 𝐴 ) ) |
5 |
2 3 4
|
sylancl |
⊢ ( 𝐴 ∈ V → ( fi ‘ 𝐴 ) ⊆ ( fi ‘ 𝒫 ∪ 𝐴 ) ) |
6 |
|
ssinss1 |
⊢ ( 𝑥 ⊆ ∪ 𝐴 → ( 𝑥 ∩ 𝑦 ) ⊆ ∪ 𝐴 ) |
7 |
|
vex |
⊢ 𝑥 ∈ V |
8 |
7
|
elpw |
⊢ ( 𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴 ) |
9 |
7
|
inex1 |
⊢ ( 𝑥 ∩ 𝑦 ) ∈ V |
10 |
9
|
elpw |
⊢ ( ( 𝑥 ∩ 𝑦 ) ∈ 𝒫 ∪ 𝐴 ↔ ( 𝑥 ∩ 𝑦 ) ⊆ ∪ 𝐴 ) |
11 |
6 8 10
|
3imtr4i |
⊢ ( 𝑥 ∈ 𝒫 ∪ 𝐴 → ( 𝑥 ∩ 𝑦 ) ∈ 𝒫 ∪ 𝐴 ) |
12 |
11
|
adantr |
⊢ ( ( 𝑥 ∈ 𝒫 ∪ 𝐴 ∧ 𝑦 ∈ 𝒫 ∪ 𝐴 ) → ( 𝑥 ∩ 𝑦 ) ∈ 𝒫 ∪ 𝐴 ) |
13 |
12
|
rgen2 |
⊢ ∀ 𝑥 ∈ 𝒫 ∪ 𝐴 ∀ 𝑦 ∈ 𝒫 ∪ 𝐴 ( 𝑥 ∩ 𝑦 ) ∈ 𝒫 ∪ 𝐴 |
14 |
|
inficl |
⊢ ( 𝒫 ∪ 𝐴 ∈ V → ( ∀ 𝑥 ∈ 𝒫 ∪ 𝐴 ∀ 𝑦 ∈ 𝒫 ∪ 𝐴 ( 𝑥 ∩ 𝑦 ) ∈ 𝒫 ∪ 𝐴 ↔ ( fi ‘ 𝒫 ∪ 𝐴 ) = 𝒫 ∪ 𝐴 ) ) |
15 |
2 14
|
syl |
⊢ ( 𝐴 ∈ V → ( ∀ 𝑥 ∈ 𝒫 ∪ 𝐴 ∀ 𝑦 ∈ 𝒫 ∪ 𝐴 ( 𝑥 ∩ 𝑦 ) ∈ 𝒫 ∪ 𝐴 ↔ ( fi ‘ 𝒫 ∪ 𝐴 ) = 𝒫 ∪ 𝐴 ) ) |
16 |
13 15
|
mpbii |
⊢ ( 𝐴 ∈ V → ( fi ‘ 𝒫 ∪ 𝐴 ) = 𝒫 ∪ 𝐴 ) |
17 |
5 16
|
sseqtrd |
⊢ ( 𝐴 ∈ V → ( fi ‘ 𝐴 ) ⊆ 𝒫 ∪ 𝐴 ) |
18 |
|
fvprc |
⊢ ( ¬ 𝐴 ∈ V → ( fi ‘ 𝐴 ) = ∅ ) |
19 |
|
0ss |
⊢ ∅ ⊆ 𝒫 ∪ 𝐴 |
20 |
18 19
|
eqsstrdi |
⊢ ( ¬ 𝐴 ∈ V → ( fi ‘ 𝐴 ) ⊆ 𝒫 ∪ 𝐴 ) |
21 |
17 20
|
pm2.61i |
⊢ ( fi ‘ 𝐴 ) ⊆ 𝒫 ∪ 𝐴 |