Step |
Hyp |
Ref |
Expression |
1 |
|
df-dju |
|- ( A |_| B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) |
2 |
|
0ex |
|- (/) e. _V |
3 |
|
relsdom |
|- Rel ~< |
4 |
3
|
brrelex2i |
|- ( 1o ~< A -> A e. _V ) |
5 |
|
xpsnen2g |
|- ( ( (/) e. _V /\ A e. _V ) -> ( { (/) } X. A ) ~~ A ) |
6 |
2 4 5
|
sylancr |
|- ( 1o ~< A -> ( { (/) } X. A ) ~~ A ) |
7 |
|
sdomen2 |
|- ( ( { (/) } X. A ) ~~ A -> ( 1o ~< ( { (/) } X. A ) <-> 1o ~< A ) ) |
8 |
6 7
|
syl |
|- ( 1o ~< A -> ( 1o ~< ( { (/) } X. A ) <-> 1o ~< A ) ) |
9 |
8
|
ibir |
|- ( 1o ~< A -> 1o ~< ( { (/) } X. A ) ) |
10 |
|
1on |
|- 1o e. On |
11 |
3
|
brrelex2i |
|- ( 1o ~< B -> B e. _V ) |
12 |
|
xpsnen2g |
|- ( ( 1o e. On /\ B e. _V ) -> ( { 1o } X. B ) ~~ B ) |
13 |
10 11 12
|
sylancr |
|- ( 1o ~< B -> ( { 1o } X. B ) ~~ B ) |
14 |
|
sdomen2 |
|- ( ( { 1o } X. B ) ~~ B -> ( 1o ~< ( { 1o } X. B ) <-> 1o ~< B ) ) |
15 |
13 14
|
syl |
|- ( 1o ~< B -> ( 1o ~< ( { 1o } X. B ) <-> 1o ~< B ) ) |
16 |
15
|
ibir |
|- ( 1o ~< B -> 1o ~< ( { 1o } X. B ) ) |
17 |
|
unxpdom |
|- ( ( 1o ~< ( { (/) } X. A ) /\ 1o ~< ( { 1o } X. B ) ) -> ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) ~<_ ( ( { (/) } X. A ) X. ( { 1o } X. B ) ) ) |
18 |
9 16 17
|
syl2an |
|- ( ( 1o ~< A /\ 1o ~< B ) -> ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) ~<_ ( ( { (/) } X. A ) X. ( { 1o } X. B ) ) ) |
19 |
1 18
|
eqbrtrid |
|- ( ( 1o ~< A /\ 1o ~< B ) -> ( A |_| B ) ~<_ ( ( { (/) } X. A ) X. ( { 1o } X. B ) ) ) |
20 |
|
xpen |
|- ( ( ( { (/) } X. A ) ~~ A /\ ( { 1o } X. B ) ~~ B ) -> ( ( { (/) } X. A ) X. ( { 1o } X. B ) ) ~~ ( A X. B ) ) |
21 |
6 13 20
|
syl2an |
|- ( ( 1o ~< A /\ 1o ~< B ) -> ( ( { (/) } X. A ) X. ( { 1o } X. B ) ) ~~ ( A X. B ) ) |
22 |
|
domentr |
|- ( ( ( A |_| B ) ~<_ ( ( { (/) } X. A ) X. ( { 1o } X. B ) ) /\ ( ( { (/) } X. A ) X. ( { 1o } X. B ) ) ~~ ( A X. B ) ) -> ( A |_| B ) ~<_ ( A X. B ) ) |
23 |
19 21 22
|
syl2anc |
|- ( ( 1o ~< A /\ 1o ~< B ) -> ( A |_| B ) ~<_ ( A X. B ) ) |