Step |
Hyp |
Ref |
Expression |
1 |
|
df-pss |
|- ( x C. U. x <-> ( x C_ U. x /\ x =/= U. x ) ) |
2 |
|
unieq |
|- ( x = (/) -> U. x = U. (/) ) |
3 |
|
uni0 |
|- U. (/) = (/) |
4 |
2 3
|
eqtr2di |
|- ( x = (/) -> (/) = U. x ) |
5 |
|
eqtr |
|- ( ( x = (/) /\ (/) = U. x ) -> x = U. x ) |
6 |
4 5
|
mpdan |
|- ( x = (/) -> x = U. x ) |
7 |
6
|
necon3i |
|- ( x =/= U. x -> x =/= (/) ) |
8 |
7
|
anim1i |
|- ( ( x =/= U. x /\ x C_ U. x ) -> ( x =/= (/) /\ x C_ U. x ) ) |
9 |
8
|
ancoms |
|- ( ( x C_ U. x /\ x =/= U. x ) -> ( x =/= (/) /\ x C_ U. x ) ) |
10 |
1 9
|
sylbi |
|- ( x C. U. x -> ( x =/= (/) /\ x C_ U. x ) ) |
11 |
10
|
eximi |
|- ( E. x x C. U. x -> E. x ( x =/= (/) /\ x C_ U. x ) ) |
12 |
|
eqid |
|- ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } ) = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } ) |
13 |
|
eqid |
|- ( rec ( ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } ) , (/) ) |` _om ) = ( rec ( ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } ) , (/) ) |` _om ) |
14 |
|
vex |
|- x e. _V |
15 |
12 13 14 14
|
inf3lem7 |
|- ( ( x =/= (/) /\ x C_ U. x ) -> _om e. _V ) |
16 |
15
|
exlimiv |
|- ( E. x ( x =/= (/) /\ x C_ U. x ) -> _om e. _V ) |
17 |
11 16
|
syl |
|- ( E. x x C. U. x -> _om e. _V ) |
18 |
|
infeq5i |
|- ( _om e. _V -> E. x x C. U. x ) |
19 |
17 18
|
impbii |
|- ( E. x x C. U. x <-> _om e. _V ) |