Step |
Hyp |
Ref |
Expression |
1 |
|
enrefg |
|- ( A e. V -> A ~~ A ) |
2 |
1
|
3ad2ant1 |
|- ( ( A e. V /\ B e. W /\ C e. X ) -> A ~~ A ) |
3 |
|
0ex |
|- (/) e. _V |
4 |
|
simp2 |
|- ( ( A e. V /\ B e. W /\ C e. X ) -> B e. W ) |
5 |
|
xpsnen2g |
|- ( ( (/) e. _V /\ B e. W ) -> ( { (/) } X. B ) ~~ B ) |
6 |
3 4 5
|
sylancr |
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { (/) } X. B ) ~~ B ) |
7 |
6
|
ensymd |
|- ( ( A e. V /\ B e. W /\ C e. X ) -> B ~~ ( { (/) } X. B ) ) |
8 |
|
xpen |
|- ( ( A ~~ A /\ B ~~ ( { (/) } X. B ) ) -> ( A X. B ) ~~ ( A X. ( { (/) } X. B ) ) ) |
9 |
2 7 8
|
syl2anc |
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. B ) ~~ ( A X. ( { (/) } X. B ) ) ) |
10 |
|
1on |
|- 1o e. On |
11 |
|
simp3 |
|- ( ( A e. V /\ B e. W /\ C e. X ) -> C e. X ) |
12 |
|
xpsnen2g |
|- ( ( 1o e. On /\ C e. X ) -> ( { 1o } X. C ) ~~ C ) |
13 |
10 11 12
|
sylancr |
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. C ) ~~ C ) |
14 |
13
|
ensymd |
|- ( ( A e. V /\ B e. W /\ C e. X ) -> C ~~ ( { 1o } X. C ) ) |
15 |
|
xpen |
|- ( ( A ~~ A /\ C ~~ ( { 1o } X. C ) ) -> ( A X. C ) ~~ ( A X. ( { 1o } X. C ) ) ) |
16 |
2 14 15
|
syl2anc |
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. C ) ~~ ( A X. ( { 1o } X. C ) ) ) |
17 |
|
xp01disjl |
|- ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) = (/) |
18 |
17
|
xpeq2i |
|- ( A X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( A X. (/) ) |
19 |
|
xpindi |
|- ( A X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( ( A X. ( { (/) } X. B ) ) i^i ( A X. ( { 1o } X. C ) ) ) |
20 |
|
xp0 |
|- ( A X. (/) ) = (/) |
21 |
18 19 20
|
3eqtr3i |
|- ( ( A X. ( { (/) } X. B ) ) i^i ( A X. ( { 1o } X. C ) ) ) = (/) |
22 |
21
|
a1i |
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A X. ( { (/) } X. B ) ) i^i ( A X. ( { 1o } X. C ) ) ) = (/) ) |
23 |
|
djuenun |
|- ( ( ( A X. B ) ~~ ( A X. ( { (/) } X. B ) ) /\ ( A X. C ) ~~ ( A X. ( { 1o } X. C ) ) /\ ( ( A X. ( { (/) } X. B ) ) i^i ( A X. ( { 1o } X. C ) ) ) = (/) ) -> ( ( A X. B ) |_| ( A X. C ) ) ~~ ( ( A X. ( { (/) } X. B ) ) u. ( A X. ( { 1o } X. C ) ) ) ) |
24 |
9 16 22 23
|
syl3anc |
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A X. B ) |_| ( A X. C ) ) ~~ ( ( A X. ( { (/) } X. B ) ) u. ( A X. ( { 1o } X. C ) ) ) ) |
25 |
|
df-dju |
|- ( B |_| C ) = ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) |
26 |
25
|
xpeq2i |
|- ( A X. ( B |_| C ) ) = ( A X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) ) |
27 |
|
xpundi |
|- ( A X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) ) = ( ( A X. ( { (/) } X. B ) ) u. ( A X. ( { 1o } X. C ) ) ) |
28 |
26 27
|
eqtri |
|- ( A X. ( B |_| C ) ) = ( ( A X. ( { (/) } X. B ) ) u. ( A X. ( { 1o } X. C ) ) ) |
29 |
24 28
|
breqtrrdi |
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A X. B ) |_| ( A X. C ) ) ~~ ( A X. ( B |_| C ) ) ) |
30 |
29
|
ensymd |
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. ( B |_| C ) ) ~~ ( ( A X. B ) |_| ( A X. C ) ) ) |