Step |
Hyp |
Ref |
Expression |
1 |
|
atne0 |
|- ( A e. HAtoms -> A =/= 0H ) |
2 |
1
|
ad2antrr |
|- ( ( ( A e. HAtoms /\ B e. HAtoms ) /\ A C_ B ) -> A =/= 0H ) |
3 |
|
atelch |
|- ( A e. HAtoms -> A e. CH ) |
4 |
|
atss |
|- ( ( A e. CH /\ B e. HAtoms ) -> ( A C_ B -> ( A = B \/ A = 0H ) ) ) |
5 |
3 4
|
sylan |
|- ( ( A e. HAtoms /\ B e. HAtoms ) -> ( A C_ B -> ( A = B \/ A = 0H ) ) ) |
6 |
5
|
imp |
|- ( ( ( A e. HAtoms /\ B e. HAtoms ) /\ A C_ B ) -> ( A = B \/ A = 0H ) ) |
7 |
6
|
ord |
|- ( ( ( A e. HAtoms /\ B e. HAtoms ) /\ A C_ B ) -> ( -. A = B -> A = 0H ) ) |
8 |
7
|
necon1ad |
|- ( ( ( A e. HAtoms /\ B e. HAtoms ) /\ A C_ B ) -> ( A =/= 0H -> A = B ) ) |
9 |
2 8
|
mpd |
|- ( ( ( A e. HAtoms /\ B e. HAtoms ) /\ A C_ B ) -> A = B ) |
10 |
9
|
ex |
|- ( ( A e. HAtoms /\ B e. HAtoms ) -> ( A C_ B -> A = B ) ) |
11 |
|
eqimss |
|- ( A = B -> A C_ B ) |
12 |
10 11
|
impbid1 |
|- ( ( A e. HAtoms /\ B e. HAtoms ) -> ( A C_ B <-> A = B ) ) |