Step |
Hyp |
Ref |
Expression |
1 |
|
helsh |
|- ~H e. SH |
2 |
|
shocel |
|- ( ~H e. SH -> ( x e. ( _|_ ` ~H ) <-> ( x e. ~H /\ A. y e. ~H ( x .ih y ) = 0 ) ) ) |
3 |
1 2
|
ax-mp |
|- ( x e. ( _|_ ` ~H ) <-> ( x e. ~H /\ A. y e. ~H ( x .ih y ) = 0 ) ) |
4 |
3
|
simprbi |
|- ( x e. ( _|_ ` ~H ) -> A. y e. ~H ( x .ih y ) = 0 ) |
5 |
|
shocss |
|- ( ~H e. SH -> ( _|_ ` ~H ) C_ ~H ) |
6 |
1 5
|
ax-mp |
|- ( _|_ ` ~H ) C_ ~H |
7 |
6
|
sseli |
|- ( x e. ( _|_ ` ~H ) -> x e. ~H ) |
8 |
|
hial0 |
|- ( x e. ~H -> ( A. y e. ~H ( x .ih y ) = 0 <-> x = 0h ) ) |
9 |
7 8
|
syl |
|- ( x e. ( _|_ ` ~H ) -> ( A. y e. ~H ( x .ih y ) = 0 <-> x = 0h ) ) |
10 |
4 9
|
mpbid |
|- ( x e. ( _|_ ` ~H ) -> x = 0h ) |
11 |
|
elch0 |
|- ( x e. 0H <-> x = 0h ) |
12 |
10 11
|
sylibr |
|- ( x e. ( _|_ ` ~H ) -> x e. 0H ) |
13 |
12
|
ssriv |
|- ( _|_ ` ~H ) C_ 0H |
14 |
|
h0elsh |
|- 0H e. SH |
15 |
|
shococss |
|- ( 0H e. SH -> 0H C_ ( _|_ ` ( _|_ ` 0H ) ) ) |
16 |
14 15
|
ax-mp |
|- 0H C_ ( _|_ ` ( _|_ ` 0H ) ) |
17 |
|
choc0 |
|- ( _|_ ` 0H ) = ~H |
18 |
17
|
fveq2i |
|- ( _|_ ` ( _|_ ` 0H ) ) = ( _|_ ` ~H ) |
19 |
16 18
|
sseqtri |
|- 0H C_ ( _|_ ` ~H ) |
20 |
13 19
|
eqssi |
|- ( _|_ ` ~H ) = 0H |