Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
|- ( ( A =/= (/) /\ A ~~ ( A |_| A ) ) -> A =/= (/) ) |
2 |
|
relen |
|- Rel ~~ |
3 |
2
|
brrelex1i |
|- ( A ~~ ( A |_| A ) -> A e. _V ) |
4 |
3
|
adantl |
|- ( ( A =/= (/) /\ A ~~ ( A |_| A ) ) -> A e. _V ) |
5 |
|
0sdomg |
|- ( A e. _V -> ( (/) ~< A <-> A =/= (/) ) ) |
6 |
4 5
|
syl |
|- ( ( A =/= (/) /\ A ~~ ( A |_| A ) ) -> ( (/) ~< A <-> A =/= (/) ) ) |
7 |
1 6
|
mpbird |
|- ( ( A =/= (/) /\ A ~~ ( A |_| A ) ) -> (/) ~< A ) |
8 |
|
0sdom1dom |
|- ( (/) ~< A <-> 1o ~<_ A ) |
9 |
7 8
|
sylib |
|- ( ( A =/= (/) /\ A ~~ ( A |_| A ) ) -> 1o ~<_ A ) |
10 |
|
djudom2 |
|- ( ( 1o ~<_ A /\ A e. _V ) -> ( A |_| 1o ) ~<_ ( A |_| A ) ) |
11 |
9 4 10
|
syl2anc |
|- ( ( A =/= (/) /\ A ~~ ( A |_| A ) ) -> ( A |_| 1o ) ~<_ ( A |_| A ) ) |
12 |
|
domen2 |
|- ( A ~~ ( A |_| A ) -> ( ( A |_| 1o ) ~<_ A <-> ( A |_| 1o ) ~<_ ( A |_| A ) ) ) |
13 |
12
|
adantl |
|- ( ( A =/= (/) /\ A ~~ ( A |_| A ) ) -> ( ( A |_| 1o ) ~<_ A <-> ( A |_| 1o ) ~<_ ( A |_| A ) ) ) |
14 |
11 13
|
mpbird |
|- ( ( A =/= (/) /\ A ~~ ( A |_| A ) ) -> ( A |_| 1o ) ~<_ A ) |
15 |
|
domnsym |
|- ( ( A |_| 1o ) ~<_ A -> -. A ~< ( A |_| 1o ) ) |
16 |
14 15
|
syl |
|- ( ( A =/= (/) /\ A ~~ ( A |_| A ) ) -> -. A ~< ( A |_| 1o ) ) |
17 |
|
isfin4p1 |
|- ( A e. Fin4 <-> A ~< ( A |_| 1o ) ) |
18 |
17
|
biimpi |
|- ( A e. Fin4 -> A ~< ( A |_| 1o ) ) |
19 |
16 18
|
nsyl3 |
|- ( A e. Fin4 -> -. ( A =/= (/) /\ A ~~ ( A |_| A ) ) ) |
20 |
|
isfin5-2 |
|- ( A e. Fin4 -> ( A e. Fin5 <-> -. ( A =/= (/) /\ A ~~ ( A |_| A ) ) ) ) |
21 |
19 20
|
mpbird |
|- ( A e. Fin4 -> A e. Fin5 ) |