Step |
Hyp |
Ref |
Expression |
1 |
|
helsh |
⊢ ℋ ∈ Sℋ |
2 |
|
shocel |
⊢ ( ℋ ∈ Sℋ → ( 𝑥 ∈ ( ⊥ ‘ ℋ ) ↔ ( 𝑥 ∈ ℋ ∧ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih 𝑦 ) = 0 ) ) ) |
3 |
1 2
|
ax-mp |
⊢ ( 𝑥 ∈ ( ⊥ ‘ ℋ ) ↔ ( 𝑥 ∈ ℋ ∧ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih 𝑦 ) = 0 ) ) |
4 |
3
|
simprbi |
⊢ ( 𝑥 ∈ ( ⊥ ‘ ℋ ) → ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih 𝑦 ) = 0 ) |
5 |
|
shocss |
⊢ ( ℋ ∈ Sℋ → ( ⊥ ‘ ℋ ) ⊆ ℋ ) |
6 |
1 5
|
ax-mp |
⊢ ( ⊥ ‘ ℋ ) ⊆ ℋ |
7 |
6
|
sseli |
⊢ ( 𝑥 ∈ ( ⊥ ‘ ℋ ) → 𝑥 ∈ ℋ ) |
8 |
|
hial0 |
⊢ ( 𝑥 ∈ ℋ → ( ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih 𝑦 ) = 0 ↔ 𝑥 = 0ℎ ) ) |
9 |
7 8
|
syl |
⊢ ( 𝑥 ∈ ( ⊥ ‘ ℋ ) → ( ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih 𝑦 ) = 0 ↔ 𝑥 = 0ℎ ) ) |
10 |
4 9
|
mpbid |
⊢ ( 𝑥 ∈ ( ⊥ ‘ ℋ ) → 𝑥 = 0ℎ ) |
11 |
|
elch0 |
⊢ ( 𝑥 ∈ 0ℋ ↔ 𝑥 = 0ℎ ) |
12 |
10 11
|
sylibr |
⊢ ( 𝑥 ∈ ( ⊥ ‘ ℋ ) → 𝑥 ∈ 0ℋ ) |
13 |
12
|
ssriv |
⊢ ( ⊥ ‘ ℋ ) ⊆ 0ℋ |
14 |
|
h0elsh |
⊢ 0ℋ ∈ Sℋ |
15 |
|
shococss |
⊢ ( 0ℋ ∈ Sℋ → 0ℋ ⊆ ( ⊥ ‘ ( ⊥ ‘ 0ℋ ) ) ) |
16 |
14 15
|
ax-mp |
⊢ 0ℋ ⊆ ( ⊥ ‘ ( ⊥ ‘ 0ℋ ) ) |
17 |
|
choc0 |
⊢ ( ⊥ ‘ 0ℋ ) = ℋ |
18 |
17
|
fveq2i |
⊢ ( ⊥ ‘ ( ⊥ ‘ 0ℋ ) ) = ( ⊥ ‘ ℋ ) |
19 |
16 18
|
sseqtri |
⊢ 0ℋ ⊆ ( ⊥ ‘ ℋ ) |
20 |
13 19
|
eqssi |
⊢ ( ⊥ ‘ ℋ ) = 0ℋ |