Step |
Hyp |
Ref |
Expression |
1 |
|
encv |
|- ( A ~~ suc M -> ( A e. _V /\ suc M e. _V ) ) |
2 |
1
|
simpld |
|- ( A ~~ suc M -> A e. _V ) |
3 |
|
breng |
|- ( ( A e. _V /\ suc M e. _V ) -> ( A ~~ suc M <-> E. f f : A -1-1-onto-> suc M ) ) |
4 |
1 3
|
syl |
|- ( A ~~ suc M -> ( A ~~ suc M <-> E. f f : A -1-1-onto-> suc M ) ) |
5 |
4
|
ibi |
|- ( A ~~ suc M -> E. f f : A -1-1-onto-> suc M ) |
6 |
|
sucidg |
|- ( M e. On -> M e. suc M ) |
7 |
|
f1ocnvdm |
|- ( ( f : A -1-1-onto-> suc M /\ M e. suc M ) -> ( `' f ` M ) e. A ) |
8 |
7
|
ancoms |
|- ( ( M e. suc M /\ f : A -1-1-onto-> suc M ) -> ( `' f ` M ) e. A ) |
9 |
6 8
|
sylan |
|- ( ( M e. On /\ f : A -1-1-onto-> suc M ) -> ( `' f ` M ) e. A ) |
10 |
9
|
adantll |
|- ( ( ( A e. _V /\ M e. On ) /\ f : A -1-1-onto-> suc M ) -> ( `' f ` M ) e. A ) |
11 |
|
vex |
|- f e. _V |
12 |
|
dif1enlem |
|- ( ( ( f e. _V /\ A e. _V /\ M e. On ) /\ f : A -1-1-onto-> suc M ) -> ( A \ { ( `' f ` M ) } ) ~~ M ) |
13 |
11 12
|
mp3anl1 |
|- ( ( ( A e. _V /\ M e. On ) /\ f : A -1-1-onto-> suc M ) -> ( A \ { ( `' f ` M ) } ) ~~ M ) |
14 |
|
sneq |
|- ( x = ( `' f ` M ) -> { x } = { ( `' f ` M ) } ) |
15 |
14
|
difeq2d |
|- ( x = ( `' f ` M ) -> ( A \ { x } ) = ( A \ { ( `' f ` M ) } ) ) |
16 |
15
|
breq1d |
|- ( x = ( `' f ` M ) -> ( ( A \ { x } ) ~~ M <-> ( A \ { ( `' f ` M ) } ) ~~ M ) ) |
17 |
16
|
rspcev |
|- ( ( ( `' f ` M ) e. A /\ ( A \ { ( `' f ` M ) } ) ~~ M ) -> E. x e. A ( A \ { x } ) ~~ M ) |
18 |
10 13 17
|
syl2anc |
|- ( ( ( A e. _V /\ M e. On ) /\ f : A -1-1-onto-> suc M ) -> E. x e. A ( A \ { x } ) ~~ M ) |
19 |
18
|
ex |
|- ( ( A e. _V /\ M e. On ) -> ( f : A -1-1-onto-> suc M -> E. x e. A ( A \ { x } ) ~~ M ) ) |
20 |
19
|
exlimdv |
|- ( ( A e. _V /\ M e. On ) -> ( E. f f : A -1-1-onto-> suc M -> E. x e. A ( A \ { x } ) ~~ M ) ) |
21 |
5 20
|
syl5 |
|- ( ( A e. _V /\ M e. On ) -> ( A ~~ suc M -> E. x e. A ( A \ { x } ) ~~ M ) ) |
22 |
2 21
|
sylan |
|- ( ( A ~~ suc M /\ M e. On ) -> ( A ~~ suc M -> E. x e. A ( A \ { x } ) ~~ M ) ) |
23 |
22
|
ancoms |
|- ( ( M e. On /\ A ~~ suc M ) -> ( A ~~ suc M -> E. x e. A ( A \ { x } ) ~~ M ) ) |
24 |
23
|
syldbl2 |
|- ( ( M e. On /\ A ~~ suc M ) -> E. x e. A ( A \ { x } ) ~~ M ) |