Step |
Hyp |
Ref |
Expression |
1 |
|
onsseleq |
|- ( ( A e. On /\ B e. On ) -> ( A C_ B <-> ( A e. B \/ A = B ) ) ) |
2 |
|
alephord |
|- ( ( A e. On /\ B e. On ) -> ( A e. B <-> ( aleph ` A ) ~< ( aleph ` B ) ) ) |
3 |
|
sdomdom |
|- ( ( aleph ` A ) ~< ( aleph ` B ) -> ( aleph ` A ) ~<_ ( aleph ` B ) ) |
4 |
2 3
|
syl6bi |
|- ( ( A e. On /\ B e. On ) -> ( A e. B -> ( aleph ` A ) ~<_ ( aleph ` B ) ) ) |
5 |
|
fvex |
|- ( aleph ` A ) e. _V |
6 |
|
fveq2 |
|- ( A = B -> ( aleph ` A ) = ( aleph ` B ) ) |
7 |
|
eqeng |
|- ( ( aleph ` A ) e. _V -> ( ( aleph ` A ) = ( aleph ` B ) -> ( aleph ` A ) ~~ ( aleph ` B ) ) ) |
8 |
5 6 7
|
mpsyl |
|- ( A = B -> ( aleph ` A ) ~~ ( aleph ` B ) ) |
9 |
8
|
a1i |
|- ( ( A e. On /\ B e. On ) -> ( A = B -> ( aleph ` A ) ~~ ( aleph ` B ) ) ) |
10 |
|
endom |
|- ( ( aleph ` A ) ~~ ( aleph ` B ) -> ( aleph ` A ) ~<_ ( aleph ` B ) ) |
11 |
9 10
|
syl6 |
|- ( ( A e. On /\ B e. On ) -> ( A = B -> ( aleph ` A ) ~<_ ( aleph ` B ) ) ) |
12 |
4 11
|
jaod |
|- ( ( A e. On /\ B e. On ) -> ( ( A e. B \/ A = B ) -> ( aleph ` A ) ~<_ ( aleph ` B ) ) ) |
13 |
1 12
|
sylbid |
|- ( ( A e. On /\ B e. On ) -> ( A C_ B -> ( aleph ` A ) ~<_ ( aleph ` B ) ) ) |
14 |
|
eloni |
|- ( B e. On -> Ord B ) |
15 |
|
eloni |
|- ( A e. On -> Ord A ) |
16 |
|
ordtri2or |
|- ( ( Ord B /\ Ord A ) -> ( B e. A \/ A C_ B ) ) |
17 |
14 15 16
|
syl2anr |
|- ( ( A e. On /\ B e. On ) -> ( B e. A \/ A C_ B ) ) |
18 |
17
|
ord |
|- ( ( A e. On /\ B e. On ) -> ( -. B e. A -> A C_ B ) ) |
19 |
18
|
con1d |
|- ( ( A e. On /\ B e. On ) -> ( -. A C_ B -> B e. A ) ) |
20 |
|
alephord |
|- ( ( B e. On /\ A e. On ) -> ( B e. A <-> ( aleph ` B ) ~< ( aleph ` A ) ) ) |
21 |
20
|
ancoms |
|- ( ( A e. On /\ B e. On ) -> ( B e. A <-> ( aleph ` B ) ~< ( aleph ` A ) ) ) |
22 |
|
sdomnen |
|- ( ( aleph ` B ) ~< ( aleph ` A ) -> -. ( aleph ` B ) ~~ ( aleph ` A ) ) |
23 |
|
sdomdom |
|- ( ( aleph ` B ) ~< ( aleph ` A ) -> ( aleph ` B ) ~<_ ( aleph ` A ) ) |
24 |
|
sbth |
|- ( ( ( aleph ` B ) ~<_ ( aleph ` A ) /\ ( aleph ` A ) ~<_ ( aleph ` B ) ) -> ( aleph ` B ) ~~ ( aleph ` A ) ) |
25 |
24
|
ex |
|- ( ( aleph ` B ) ~<_ ( aleph ` A ) -> ( ( aleph ` A ) ~<_ ( aleph ` B ) -> ( aleph ` B ) ~~ ( aleph ` A ) ) ) |
26 |
23 25
|
syl |
|- ( ( aleph ` B ) ~< ( aleph ` A ) -> ( ( aleph ` A ) ~<_ ( aleph ` B ) -> ( aleph ` B ) ~~ ( aleph ` A ) ) ) |
27 |
22 26
|
mtod |
|- ( ( aleph ` B ) ~< ( aleph ` A ) -> -. ( aleph ` A ) ~<_ ( aleph ` B ) ) |
28 |
21 27
|
syl6bi |
|- ( ( A e. On /\ B e. On ) -> ( B e. A -> -. ( aleph ` A ) ~<_ ( aleph ` B ) ) ) |
29 |
19 28
|
syld |
|- ( ( A e. On /\ B e. On ) -> ( -. A C_ B -> -. ( aleph ` A ) ~<_ ( aleph ` B ) ) ) |
30 |
13 29
|
impcon4bid |
|- ( ( A e. On /\ B e. On ) -> ( A C_ B <-> ( aleph ` A ) ~<_ ( aleph ` B ) ) ) |