Step |
Hyp |
Ref |
Expression |
1 |
|
2ne0 |
|- 2 =/= 0 |
2 |
1
|
neii |
|- -. 2 = 0 |
3 |
|
eqeq1 |
|- ( ( # ` x ) = 2 -> ( ( # ` x ) = 0 <-> 2 = 0 ) ) |
4 |
2 3
|
mtbiri |
|- ( ( # ` x ) = 2 -> -. ( # ` x ) = 0 ) |
5 |
|
hasheq0 |
|- ( x e. _V -> ( ( # ` x ) = 0 <-> x = (/) ) ) |
6 |
5
|
bicomd |
|- ( x e. _V -> ( x = (/) <-> ( # ` x ) = 0 ) ) |
7 |
6
|
necon3abid |
|- ( x e. _V -> ( x =/= (/) <-> -. ( # ` x ) = 0 ) ) |
8 |
7
|
elv |
|- ( x =/= (/) <-> -. ( # ` x ) = 0 ) |
9 |
4 8
|
sylibr |
|- ( ( # ` x ) = 2 -> x =/= (/) ) |
10 |
9
|
biantrud |
|- ( ( # ` x ) = 2 -> ( x e. ~P A <-> ( x e. ~P A /\ x =/= (/) ) ) ) |
11 |
|
eldifsn |
|- ( x e. ( ~P A \ { (/) } ) <-> ( x e. ~P A /\ x =/= (/) ) ) |
12 |
10 11
|
bitr4di |
|- ( ( # ` x ) = 2 -> ( x e. ~P A <-> x e. ( ~P A \ { (/) } ) ) ) |
13 |
12
|
pm5.32ri |
|- ( ( x e. ~P A /\ ( # ` x ) = 2 ) <-> ( x e. ( ~P A \ { (/) } ) /\ ( # ` x ) = 2 ) ) |
14 |
13
|
abbii |
|- { x | ( x e. ~P A /\ ( # ` x ) = 2 ) } = { x | ( x e. ( ~P A \ { (/) } ) /\ ( # ` x ) = 2 ) } |
15 |
|
df-rab |
|- { x e. ~P A | ( # ` x ) = 2 } = { x | ( x e. ~P A /\ ( # ` x ) = 2 ) } |
16 |
|
df-rab |
|- { x e. ( ~P A \ { (/) } ) | ( # ` x ) = 2 } = { x | ( x e. ( ~P A \ { (/) } ) /\ ( # ` x ) = 2 ) } |
17 |
14 15 16
|
3eqtr4ri |
|- { x e. ( ~P A \ { (/) } ) | ( # ` x ) = 2 } = { x e. ~P A | ( # ` x ) = 2 } |