Step |
Hyp |
Ref |
Expression |
1 |
|
ocss |
⊢ ( 𝐴 ⊆ ℋ → ( ⊥ ‘ 𝐴 ) ⊆ ℋ ) |
2 |
|
sshjval |
⊢ ( ( 𝐴 ⊆ ℋ ∧ ( ⊥ ‘ 𝐴 ) ⊆ ℋ ) → ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ) ) |
3 |
1 2
|
mpdan |
⊢ ( 𝐴 ⊆ ℋ → ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ) ) |
4 |
|
ssun1 |
⊢ 𝐴 ⊆ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) |
5 |
1
|
ancli |
⊢ ( 𝐴 ⊆ ℋ → ( 𝐴 ⊆ ℋ ∧ ( ⊥ ‘ 𝐴 ) ⊆ ℋ ) ) |
6 |
|
unss |
⊢ ( ( 𝐴 ⊆ ℋ ∧ ( ⊥ ‘ 𝐴 ) ⊆ ℋ ) ↔ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ⊆ ℋ ) |
7 |
5 6
|
sylib |
⊢ ( 𝐴 ⊆ ℋ → ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ⊆ ℋ ) |
8 |
|
occon |
⊢ ( ( 𝐴 ⊆ ℋ ∧ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ⊆ ℋ ) → ( 𝐴 ⊆ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) → ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ⊆ ( ⊥ ‘ 𝐴 ) ) ) |
9 |
7 8
|
mpdan |
⊢ ( 𝐴 ⊆ ℋ → ( 𝐴 ⊆ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) → ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ⊆ ( ⊥ ‘ 𝐴 ) ) ) |
10 |
4 9
|
mpi |
⊢ ( 𝐴 ⊆ ℋ → ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ⊆ ( ⊥ ‘ 𝐴 ) ) |
11 |
|
ssun2 |
⊢ ( ⊥ ‘ 𝐴 ) ⊆ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) |
12 |
|
occon |
⊢ ( ( ( ⊥ ‘ 𝐴 ) ⊆ ℋ ∧ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ⊆ ℋ ) → ( ( ⊥ ‘ 𝐴 ) ⊆ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) → ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ) ) |
13 |
1 7 12
|
syl2anc |
⊢ ( 𝐴 ⊆ ℋ → ( ( ⊥ ‘ 𝐴 ) ⊆ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) → ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ) ) |
14 |
11 13
|
mpi |
⊢ ( 𝐴 ⊆ ℋ → ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ) |
15 |
10 14
|
ssind |
⊢ ( 𝐴 ⊆ ℋ → ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ⊆ ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ) ) |
16 |
|
ocsh |
⊢ ( 𝐴 ⊆ ℋ → ( ⊥ ‘ 𝐴 ) ∈ Sℋ ) |
17 |
|
ocin |
⊢ ( ( ⊥ ‘ 𝐴 ) ∈ Sℋ → ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ) = 0ℋ ) |
18 |
16 17
|
syl |
⊢ ( 𝐴 ⊆ ℋ → ( ( ⊥ ‘ 𝐴 ) ∩ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ) = 0ℋ ) |
19 |
15 18
|
sseqtrd |
⊢ ( 𝐴 ⊆ ℋ → ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ⊆ 0ℋ ) |
20 |
|
ocsh |
⊢ ( ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ⊆ ℋ → ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ∈ Sℋ ) |
21 |
|
sh0le |
⊢ ( ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ∈ Sℋ → 0ℋ ⊆ ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ) |
22 |
7 20 21
|
3syl |
⊢ ( 𝐴 ⊆ ℋ → 0ℋ ⊆ ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ) |
23 |
19 22
|
eqssd |
⊢ ( 𝐴 ⊆ ℋ → ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) = 0ℋ ) |
24 |
23
|
fveq2d |
⊢ ( 𝐴 ⊆ ℋ → ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ) = ( ⊥ ‘ 0ℋ ) ) |
25 |
|
choc0 |
⊢ ( ⊥ ‘ 0ℋ ) = ℋ |
26 |
24 25
|
eqtrdi |
⊢ ( 𝐴 ⊆ ℋ → ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ ( ⊥ ‘ 𝐴 ) ) ) ) = ℋ ) |
27 |
3 26
|
eqtrd |
⊢ ( 𝐴 ⊆ ℋ → ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ℋ ) |