Step |
Hyp |
Ref |
Expression |
1 |
|
rankvalg |
|- ( A e. B -> ( rank ` A ) = |^| { x e. On | A e. ( R1 ` suc x ) } ) |
2 |
|
r1suc |
|- ( x e. On -> ( R1 ` suc x ) = ~P ( R1 ` x ) ) |
3 |
2
|
eleq2d |
|- ( x e. On -> ( A e. ( R1 ` suc x ) <-> A e. ~P ( R1 ` x ) ) ) |
4 |
|
fvex |
|- ( R1 ` x ) e. _V |
5 |
4
|
elpw2 |
|- ( A e. ~P ( R1 ` x ) <-> A C_ ( R1 ` x ) ) |
6 |
3 5
|
syl6bb |
|- ( x e. On -> ( A e. ( R1 ` suc x ) <-> A C_ ( R1 ` x ) ) ) |
7 |
6
|
rabbiia |
|- { x e. On | A e. ( R1 ` suc x ) } = { x e. On | A C_ ( R1 ` x ) } |
8 |
7
|
inteqi |
|- |^| { x e. On | A e. ( R1 ` suc x ) } = |^| { x e. On | A C_ ( R1 ` x ) } |
9 |
1 8
|
eqtrdi |
|- ( A e. B -> ( rank ` A ) = |^| { x e. On | A C_ ( R1 ` x ) } ) |