| Step |
Hyp |
Ref |
Expression |
| 1 |
|
neq0 |
|- ( -. A = (/) <-> E. x x e. A ) |
| 2 |
|
ssel |
|- ( A C_ { B } -> ( x e. A -> x e. { B } ) ) |
| 3 |
|
elsni |
|- ( x e. { B } -> x = B ) |
| 4 |
2 3
|
syl6 |
|- ( A C_ { B } -> ( x e. A -> x = B ) ) |
| 5 |
|
eleq1 |
|- ( x = B -> ( x e. A <-> B e. A ) ) |
| 6 |
4 5
|
syl6 |
|- ( A C_ { B } -> ( x e. A -> ( x e. A <-> B e. A ) ) ) |
| 7 |
6
|
ibd |
|- ( A C_ { B } -> ( x e. A -> B e. A ) ) |
| 8 |
7
|
exlimdv |
|- ( A C_ { B } -> ( E. x x e. A -> B e. A ) ) |
| 9 |
1 8
|
biimtrid |
|- ( A C_ { B } -> ( -. A = (/) -> B e. A ) ) |
| 10 |
|
snssi |
|- ( B e. A -> { B } C_ A ) |
| 11 |
9 10
|
syl6 |
|- ( A C_ { B } -> ( -. A = (/) -> { B } C_ A ) ) |
| 12 |
11
|
anc2li |
|- ( A C_ { B } -> ( -. A = (/) -> ( A C_ { B } /\ { B } C_ A ) ) ) |
| 13 |
|
eqss |
|- ( A = { B } <-> ( A C_ { B } /\ { B } C_ A ) ) |
| 14 |
12 13
|
imbitrrdi |
|- ( A C_ { B } -> ( -. A = (/) -> A = { B } ) ) |
| 15 |
14
|
orrd |
|- ( A C_ { B } -> ( A = (/) \/ A = { B } ) ) |
| 16 |
|
0ss |
|- (/) C_ { B } |
| 17 |
|
sseq1 |
|- ( A = (/) -> ( A C_ { B } <-> (/) C_ { B } ) ) |
| 18 |
16 17
|
mpbiri |
|- ( A = (/) -> A C_ { B } ) |
| 19 |
|
eqimss |
|- ( A = { B } -> A C_ { B } ) |
| 20 |
18 19
|
jaoi |
|- ( ( A = (/) \/ A = { B } ) -> A C_ { B } ) |
| 21 |
15 20
|
impbii |
|- ( A C_ { B } <-> ( A = (/) \/ A = { B } ) ) |