| Step |
Hyp |
Ref |
Expression |
| 1 |
|
n0 |
|- ( A =/= (/) <-> E. y y e. A ) |
| 2 |
|
vex |
|- y e. _V |
| 3 |
2
|
snss |
|- ( y e. A <-> { y } C_ A ) |
| 4 |
2
|
snnz |
|- { y } =/= (/) |
| 5 |
|
vsnex |
|- { y } e. _V |
| 6 |
|
sseq1 |
|- ( x = { y } -> ( x C_ A <-> { y } C_ A ) ) |
| 7 |
|
neeq1 |
|- ( x = { y } -> ( x =/= (/) <-> { y } =/= (/) ) ) |
| 8 |
6 7
|
anbi12d |
|- ( x = { y } -> ( ( x C_ A /\ x =/= (/) ) <-> ( { y } C_ A /\ { y } =/= (/) ) ) ) |
| 9 |
5 8
|
spcev |
|- ( ( { y } C_ A /\ { y } =/= (/) ) -> E. x ( x C_ A /\ x =/= (/) ) ) |
| 10 |
4 9
|
mpan2 |
|- ( { y } C_ A -> E. x ( x C_ A /\ x =/= (/) ) ) |
| 11 |
3 10
|
sylbi |
|- ( y e. A -> E. x ( x C_ A /\ x =/= (/) ) ) |
| 12 |
11
|
exlimiv |
|- ( E. y y e. A -> E. x ( x C_ A /\ x =/= (/) ) ) |
| 13 |
1 12
|
sylbi |
|- ( A =/= (/) -> E. x ( x C_ A /\ x =/= (/) ) ) |