Step |
Hyp |
Ref |
Expression |
1 |
|
orduninsuc |
|- ( Ord A -> ( A = U. A <-> -. E. x e. On A = suc x ) ) |
2 |
|
ralnex |
|- ( A. x e. On -. A = suc x <-> -. E. x e. On A = suc x ) |
3 |
|
suceloni |
|- ( x e. On -> suc x e. On ) |
4 |
|
eloni |
|- ( suc x e. On -> Ord suc x ) |
5 |
3 4
|
syl |
|- ( x e. On -> Ord suc x ) |
6 |
|
ordtri3 |
|- ( ( Ord A /\ Ord suc x ) -> ( A = suc x <-> -. ( A e. suc x \/ suc x e. A ) ) ) |
7 |
5 6
|
sylan2 |
|- ( ( Ord A /\ x e. On ) -> ( A = suc x <-> -. ( A e. suc x \/ suc x e. A ) ) ) |
8 |
7
|
con2bid |
|- ( ( Ord A /\ x e. On ) -> ( ( A e. suc x \/ suc x e. A ) <-> -. A = suc x ) ) |
9 |
|
onnbtwn |
|- ( x e. On -> -. ( x e. A /\ A e. suc x ) ) |
10 |
|
imnan |
|- ( ( x e. A -> -. A e. suc x ) <-> -. ( x e. A /\ A e. suc x ) ) |
11 |
9 10
|
sylibr |
|- ( x e. On -> ( x e. A -> -. A e. suc x ) ) |
12 |
11
|
con2d |
|- ( x e. On -> ( A e. suc x -> -. x e. A ) ) |
13 |
|
pm2.21 |
|- ( -. x e. A -> ( x e. A -> suc x e. A ) ) |
14 |
12 13
|
syl6 |
|- ( x e. On -> ( A e. suc x -> ( x e. A -> suc x e. A ) ) ) |
15 |
14
|
adantl |
|- ( ( Ord A /\ x e. On ) -> ( A e. suc x -> ( x e. A -> suc x e. A ) ) ) |
16 |
|
ax-1 |
|- ( suc x e. A -> ( x e. A -> suc x e. A ) ) |
17 |
16
|
a1i |
|- ( ( Ord A /\ x e. On ) -> ( suc x e. A -> ( x e. A -> suc x e. A ) ) ) |
18 |
15 17
|
jaod |
|- ( ( Ord A /\ x e. On ) -> ( ( A e. suc x \/ suc x e. A ) -> ( x e. A -> suc x e. A ) ) ) |
19 |
|
eloni |
|- ( x e. On -> Ord x ) |
20 |
|
ordtri2or |
|- ( ( Ord x /\ Ord A ) -> ( x e. A \/ A C_ x ) ) |
21 |
19 20
|
sylan |
|- ( ( x e. On /\ Ord A ) -> ( x e. A \/ A C_ x ) ) |
22 |
21
|
ancoms |
|- ( ( Ord A /\ x e. On ) -> ( x e. A \/ A C_ x ) ) |
23 |
22
|
orcomd |
|- ( ( Ord A /\ x e. On ) -> ( A C_ x \/ x e. A ) ) |
24 |
23
|
adantr |
|- ( ( ( Ord A /\ x e. On ) /\ ( x e. A -> suc x e. A ) ) -> ( A C_ x \/ x e. A ) ) |
25 |
|
ordsssuc2 |
|- ( ( Ord A /\ x e. On ) -> ( A C_ x <-> A e. suc x ) ) |
26 |
25
|
biimpd |
|- ( ( Ord A /\ x e. On ) -> ( A C_ x -> A e. suc x ) ) |
27 |
26
|
adantr |
|- ( ( ( Ord A /\ x e. On ) /\ ( x e. A -> suc x e. A ) ) -> ( A C_ x -> A e. suc x ) ) |
28 |
|
simpr |
|- ( ( ( Ord A /\ x e. On ) /\ ( x e. A -> suc x e. A ) ) -> ( x e. A -> suc x e. A ) ) |
29 |
27 28
|
orim12d |
|- ( ( ( Ord A /\ x e. On ) /\ ( x e. A -> suc x e. A ) ) -> ( ( A C_ x \/ x e. A ) -> ( A e. suc x \/ suc x e. A ) ) ) |
30 |
24 29
|
mpd |
|- ( ( ( Ord A /\ x e. On ) /\ ( x e. A -> suc x e. A ) ) -> ( A e. suc x \/ suc x e. A ) ) |
31 |
30
|
ex |
|- ( ( Ord A /\ x e. On ) -> ( ( x e. A -> suc x e. A ) -> ( A e. suc x \/ suc x e. A ) ) ) |
32 |
18 31
|
impbid |
|- ( ( Ord A /\ x e. On ) -> ( ( A e. suc x \/ suc x e. A ) <-> ( x e. A -> suc x e. A ) ) ) |
33 |
8 32
|
bitr3d |
|- ( ( Ord A /\ x e. On ) -> ( -. A = suc x <-> ( x e. A -> suc x e. A ) ) ) |
34 |
33
|
pm5.74da |
|- ( Ord A -> ( ( x e. On -> -. A = suc x ) <-> ( x e. On -> ( x e. A -> suc x e. A ) ) ) ) |
35 |
|
impexp |
|- ( ( ( x e. On /\ x e. A ) -> suc x e. A ) <-> ( x e. On -> ( x e. A -> suc x e. A ) ) ) |
36 |
|
simpr |
|- ( ( x e. On /\ x e. A ) -> x e. A ) |
37 |
|
ordelon |
|- ( ( Ord A /\ x e. A ) -> x e. On ) |
38 |
37
|
ex |
|- ( Ord A -> ( x e. A -> x e. On ) ) |
39 |
38
|
ancrd |
|- ( Ord A -> ( x e. A -> ( x e. On /\ x e. A ) ) ) |
40 |
36 39
|
impbid2 |
|- ( Ord A -> ( ( x e. On /\ x e. A ) <-> x e. A ) ) |
41 |
40
|
imbi1d |
|- ( Ord A -> ( ( ( x e. On /\ x e. A ) -> suc x e. A ) <-> ( x e. A -> suc x e. A ) ) ) |
42 |
35 41
|
bitr3id |
|- ( Ord A -> ( ( x e. On -> ( x e. A -> suc x e. A ) ) <-> ( x e. A -> suc x e. A ) ) ) |
43 |
34 42
|
bitrd |
|- ( Ord A -> ( ( x e. On -> -. A = suc x ) <-> ( x e. A -> suc x e. A ) ) ) |
44 |
43
|
ralbidv2 |
|- ( Ord A -> ( A. x e. On -. A = suc x <-> A. x e. A suc x e. A ) ) |
45 |
2 44
|
bitr3id |
|- ( Ord A -> ( -. E. x e. On A = suc x <-> A. x e. A suc x e. A ) ) |
46 |
1 45
|
bitrd |
|- ( Ord A -> ( A = U. A <-> A. x e. A suc x e. A ) ) |