Step |
Hyp |
Ref |
Expression |
1 |
|
df-dju |
|- ( A |_| B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) |
2 |
|
0elon |
|- (/) e. On |
3 |
|
relsdom |
|- Rel ~< |
4 |
3
|
brrelex1i |
|- ( A ~< _om -> A e. _V ) |
5 |
|
xpsnen2g |
|- ( ( (/) e. On /\ A e. _V ) -> ( { (/) } X. A ) ~~ A ) |
6 |
2 4 5
|
sylancr |
|- ( A ~< _om -> ( { (/) } X. A ) ~~ A ) |
7 |
|
sdomen1 |
|- ( ( { (/) } X. A ) ~~ A -> ( ( { (/) } X. A ) ~< _om <-> A ~< _om ) ) |
8 |
6 7
|
syl |
|- ( A ~< _om -> ( ( { (/) } X. A ) ~< _om <-> A ~< _om ) ) |
9 |
8
|
ibir |
|- ( A ~< _om -> ( { (/) } X. A ) ~< _om ) |
10 |
|
1on |
|- 1o e. On |
11 |
3
|
brrelex1i |
|- ( B ~< _om -> B e. _V ) |
12 |
|
xpsnen2g |
|- ( ( 1o e. On /\ B e. _V ) -> ( { 1o } X. B ) ~~ B ) |
13 |
10 11 12
|
sylancr |
|- ( B ~< _om -> ( { 1o } X. B ) ~~ B ) |
14 |
|
sdomen1 |
|- ( ( { 1o } X. B ) ~~ B -> ( ( { 1o } X. B ) ~< _om <-> B ~< _om ) ) |
15 |
13 14
|
syl |
|- ( B ~< _om -> ( ( { 1o } X. B ) ~< _om <-> B ~< _om ) ) |
16 |
15
|
ibir |
|- ( B ~< _om -> ( { 1o } X. B ) ~< _om ) |
17 |
|
unfi2 |
|- ( ( ( { (/) } X. A ) ~< _om /\ ( { 1o } X. B ) ~< _om ) -> ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) ~< _om ) |
18 |
9 16 17
|
syl2an |
|- ( ( A ~< _om /\ B ~< _om ) -> ( ( { (/) } X. A ) u. ( { 1o } X. B ) ) ~< _om ) |
19 |
1 18
|
eqbrtrid |
|- ( ( A ~< _om /\ B ~< _om ) -> ( A |_| B ) ~< _om ) |