Step |
Hyp |
Ref |
Expression |
1 |
|
brdomi |
|- ( A ~<_ B -> E. f f : A -1-1-> B ) |
2 |
1
|
3ad2ant3 |
|- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> E. f f : A -1-1-> B ) |
3 |
|
simp2 |
|- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> B C_ C ) |
4 |
|
reldom |
|- Rel ~<_ |
5 |
4
|
brrelex1i |
|- ( A ~<_ B -> A e. _V ) |
6 |
5
|
3ad2ant3 |
|- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> A e. _V ) |
7 |
|
simp1 |
|- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> C e. V ) |
8 |
3 6 7
|
jca32 |
|- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> ( B C_ C /\ ( A e. _V /\ C e. V ) ) ) |
9 |
|
f1ss |
|- ( ( f : A -1-1-> B /\ B C_ C ) -> f : A -1-1-> C ) |
10 |
|
vex |
|- f e. _V |
11 |
|
f1dom4g |
|- ( ( ( f e. _V /\ A e. _V /\ C e. V ) /\ f : A -1-1-> C ) -> A ~<_ C ) |
12 |
10 11
|
mp3anl1 |
|- ( ( ( A e. _V /\ C e. V ) /\ f : A -1-1-> C ) -> A ~<_ C ) |
13 |
12
|
ancoms |
|- ( ( f : A -1-1-> C /\ ( A e. _V /\ C e. V ) ) -> A ~<_ C ) |
14 |
9 13
|
sylan |
|- ( ( ( f : A -1-1-> B /\ B C_ C ) /\ ( A e. _V /\ C e. V ) ) -> A ~<_ C ) |
15 |
14
|
expl |
|- ( f : A -1-1-> B -> ( ( B C_ C /\ ( A e. _V /\ C e. V ) ) -> A ~<_ C ) ) |
16 |
15
|
exlimiv |
|- ( E. f f : A -1-1-> B -> ( ( B C_ C /\ ( A e. _V /\ C e. V ) ) -> A ~<_ C ) ) |
17 |
2 8 16
|
sylc |
|- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> A ~<_ C ) |