Step |
Hyp |
Ref |
Expression |
1 |
|
ordom |
|- Ord _om |
2 |
|
ordelss |
|- ( ( Ord _om /\ A e. _om ) -> A C_ _om ) |
3 |
1 2
|
mpan |
|- ( A e. _om -> A C_ _om ) |
4 |
3
|
adantr |
|- ( ( A e. _om /\ _om e. _V ) -> A C_ _om ) |
5 |
|
nnfi |
|- ( A e. _om -> A e. Fin ) |
6 |
|
ssdomfi2 |
|- ( ( A e. Fin /\ _om e. _V /\ A C_ _om ) -> A ~<_ _om ) |
7 |
5 6
|
syl3an1 |
|- ( ( A e. _om /\ _om e. _V /\ A C_ _om ) -> A ~<_ _om ) |
8 |
4 7
|
mpd3an3 |
|- ( ( A e. _om /\ _om e. _V ) -> A ~<_ _om ) |
9 |
8
|
ancoms |
|- ( ( _om e. _V /\ A e. _om ) -> A ~<_ _om ) |
10 |
|
ominf |
|- -. _om e. Fin |
11 |
|
ensymfib |
|- ( A e. Fin -> ( A ~~ _om <-> _om ~~ A ) ) |
12 |
5 11
|
syl |
|- ( A e. _om -> ( A ~~ _om <-> _om ~~ A ) ) |
13 |
|
breq2 |
|- ( x = A -> ( _om ~~ x <-> _om ~~ A ) ) |
14 |
13
|
rspcev |
|- ( ( A e. _om /\ _om ~~ A ) -> E. x e. _om _om ~~ x ) |
15 |
|
isfi |
|- ( _om e. Fin <-> E. x e. _om _om ~~ x ) |
16 |
14 15
|
sylibr |
|- ( ( A e. _om /\ _om ~~ A ) -> _om e. Fin ) |
17 |
16
|
ex |
|- ( A e. _om -> ( _om ~~ A -> _om e. Fin ) ) |
18 |
12 17
|
sylbid |
|- ( A e. _om -> ( A ~~ _om -> _om e. Fin ) ) |
19 |
10 18
|
mtoi |
|- ( A e. _om -> -. A ~~ _om ) |
20 |
19
|
adantl |
|- ( ( _om e. _V /\ A e. _om ) -> -. A ~~ _om ) |
21 |
|
brsdom |
|- ( A ~< _om <-> ( A ~<_ _om /\ -. A ~~ _om ) ) |
22 |
9 20 21
|
sylanbrc |
|- ( ( _om e. _V /\ A e. _om ) -> A ~< _om ) |