Step |
Hyp |
Ref |
Expression |
1 |
|
3ancomb |
|- ( ( Ord A /\ -. Lim A /\ A =/= (/) ) <-> ( Ord A /\ A =/= (/) /\ -. Lim A ) ) |
2 |
|
df-3an |
|- ( ( Ord A /\ A =/= (/) /\ -. Lim A ) <-> ( ( Ord A /\ A =/= (/) ) /\ -. Lim A ) ) |
3 |
|
df-ne |
|- ( A =/= (/) <-> -. A = (/) ) |
4 |
3
|
anbi2i |
|- ( ( Ord A /\ A =/= (/) ) <-> ( Ord A /\ -. A = (/) ) ) |
5 |
4
|
imbi1i |
|- ( ( ( Ord A /\ A =/= (/) ) -> E. x e. On A = suc x ) <-> ( ( Ord A /\ -. A = (/) ) -> E. x e. On A = suc x ) ) |
6 |
|
pm5.6 |
|- ( ( ( Ord A /\ -. A = (/) ) -> E. x e. On A = suc x ) <-> ( Ord A -> ( A = (/) \/ E. x e. On A = suc x ) ) ) |
7 |
|
iman |
|- ( ( Ord A -> ( A = (/) \/ E. x e. On A = suc x ) ) <-> -. ( Ord A /\ -. ( A = (/) \/ E. x e. On A = suc x ) ) ) |
8 |
5 6 7
|
3bitrri |
|- ( -. ( Ord A /\ -. ( A = (/) \/ E. x e. On A = suc x ) ) <-> ( ( Ord A /\ A =/= (/) ) -> E. x e. On A = suc x ) ) |
9 |
|
dflim3 |
|- ( Lim A <-> ( Ord A /\ -. ( A = (/) \/ E. x e. On A = suc x ) ) ) |
10 |
8 9
|
xchnxbir |
|- ( -. Lim A <-> ( ( Ord A /\ A =/= (/) ) -> E. x e. On A = suc x ) ) |
11 |
10
|
anbi2i |
|- ( ( ( Ord A /\ A =/= (/) ) /\ -. Lim A ) <-> ( ( Ord A /\ A =/= (/) ) /\ ( ( Ord A /\ A =/= (/) ) -> E. x e. On A = suc x ) ) ) |
12 |
1 2 11
|
3bitri |
|- ( ( Ord A /\ -. Lim A /\ A =/= (/) ) <-> ( ( Ord A /\ A =/= (/) ) /\ ( ( Ord A /\ A =/= (/) ) -> E. x e. On A = suc x ) ) ) |
13 |
|
pm3.35 |
|- ( ( ( Ord A /\ A =/= (/) ) /\ ( ( Ord A /\ A =/= (/) ) -> E. x e. On A = suc x ) ) -> E. x e. On A = suc x ) |
14 |
12 13
|
sylbi |
|- ( ( Ord A /\ -. Lim A /\ A =/= (/) ) -> E. x e. On A = suc x ) |
15 |
|
eloni |
|- ( x e. On -> Ord x ) |
16 |
|
ordsuc |
|- ( Ord x <-> Ord suc x ) |
17 |
15 16
|
sylib |
|- ( x e. On -> Ord suc x ) |
18 |
|
nlimsucg |
|- ( x e. On -> -. Lim suc x ) |
19 |
|
nsuceq0 |
|- suc x =/= (/) |
20 |
19
|
a1i |
|- ( x e. On -> suc x =/= (/) ) |
21 |
17 18 20
|
3jca |
|- ( x e. On -> ( Ord suc x /\ -. Lim suc x /\ suc x =/= (/) ) ) |
22 |
|
ordeq |
|- ( A = suc x -> ( Ord A <-> Ord suc x ) ) |
23 |
|
limeq |
|- ( A = suc x -> ( Lim A <-> Lim suc x ) ) |
24 |
23
|
notbid |
|- ( A = suc x -> ( -. Lim A <-> -. Lim suc x ) ) |
25 |
|
neeq1 |
|- ( A = suc x -> ( A =/= (/) <-> suc x =/= (/) ) ) |
26 |
22 24 25
|
3anbi123d |
|- ( A = suc x -> ( ( Ord A /\ -. Lim A /\ A =/= (/) ) <-> ( Ord suc x /\ -. Lim suc x /\ suc x =/= (/) ) ) ) |
27 |
21 26
|
syl5ibrcom |
|- ( x e. On -> ( A = suc x -> ( Ord A /\ -. Lim A /\ A =/= (/) ) ) ) |
28 |
27
|
rexlimiv |
|- ( E. x e. On A = suc x -> ( Ord A /\ -. Lim A /\ A =/= (/) ) ) |
29 |
14 28
|
impbii |
|- ( ( Ord A /\ -. Lim A /\ A =/= (/) ) <-> E. x e. On A = suc x ) |