Step |
Hyp |
Ref |
Expression |
1 |
|
brcup.1 |
|- A e. _V |
2 |
|
brcup.2 |
|- B e. _V |
3 |
|
brcup.3 |
|- C e. _V |
4 |
|
opex |
|- <. A , B >. e. _V |
5 |
|
df-cup |
|- Cup = ( ( ( _V X. _V ) X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( ( `' 1st o. _E ) u. ( `' 2nd o. _E ) ) (x) _V ) ) ) |
6 |
1 2
|
opelvv |
|- <. A , B >. e. ( _V X. _V ) |
7 |
|
brxp |
|- ( <. A , B >. ( ( _V X. _V ) X. _V ) C <-> ( <. A , B >. e. ( _V X. _V ) /\ C e. _V ) ) |
8 |
6 3 7
|
mpbir2an |
|- <. A , B >. ( ( _V X. _V ) X. _V ) C |
9 |
|
epel |
|- ( x _E y <-> x e. y ) |
10 |
|
vex |
|- y e. _V |
11 |
10 4
|
brcnv |
|- ( y `' 1st <. A , B >. <-> <. A , B >. 1st y ) |
12 |
1 2
|
br1steq |
|- ( <. A , B >. 1st y <-> y = A ) |
13 |
11 12
|
bitri |
|- ( y `' 1st <. A , B >. <-> y = A ) |
14 |
9 13
|
anbi12ci |
|- ( ( x _E y /\ y `' 1st <. A , B >. ) <-> ( y = A /\ x e. y ) ) |
15 |
14
|
exbii |
|- ( E. y ( x _E y /\ y `' 1st <. A , B >. ) <-> E. y ( y = A /\ x e. y ) ) |
16 |
|
vex |
|- x e. _V |
17 |
16 4
|
brco |
|- ( x ( `' 1st o. _E ) <. A , B >. <-> E. y ( x _E y /\ y `' 1st <. A , B >. ) ) |
18 |
1
|
clel3 |
|- ( x e. A <-> E. y ( y = A /\ x e. y ) ) |
19 |
15 17 18
|
3bitr4i |
|- ( x ( `' 1st o. _E ) <. A , B >. <-> x e. A ) |
20 |
10 4
|
brcnv |
|- ( y `' 2nd <. A , B >. <-> <. A , B >. 2nd y ) |
21 |
1 2
|
br2ndeq |
|- ( <. A , B >. 2nd y <-> y = B ) |
22 |
20 21
|
bitri |
|- ( y `' 2nd <. A , B >. <-> y = B ) |
23 |
9 22
|
anbi12ci |
|- ( ( x _E y /\ y `' 2nd <. A , B >. ) <-> ( y = B /\ x e. y ) ) |
24 |
23
|
exbii |
|- ( E. y ( x _E y /\ y `' 2nd <. A , B >. ) <-> E. y ( y = B /\ x e. y ) ) |
25 |
16 4
|
brco |
|- ( x ( `' 2nd o. _E ) <. A , B >. <-> E. y ( x _E y /\ y `' 2nd <. A , B >. ) ) |
26 |
2
|
clel3 |
|- ( x e. B <-> E. y ( y = B /\ x e. y ) ) |
27 |
24 25 26
|
3bitr4i |
|- ( x ( `' 2nd o. _E ) <. A , B >. <-> x e. B ) |
28 |
19 27
|
orbi12i |
|- ( ( x ( `' 1st o. _E ) <. A , B >. \/ x ( `' 2nd o. _E ) <. A , B >. ) <-> ( x e. A \/ x e. B ) ) |
29 |
|
brun |
|- ( x ( ( `' 1st o. _E ) u. ( `' 2nd o. _E ) ) <. A , B >. <-> ( x ( `' 1st o. _E ) <. A , B >. \/ x ( `' 2nd o. _E ) <. A , B >. ) ) |
30 |
|
elun |
|- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) ) |
31 |
28 29 30
|
3bitr4ri |
|- ( x e. ( A u. B ) <-> x ( ( `' 1st o. _E ) u. ( `' 2nd o. _E ) ) <. A , B >. ) |
32 |
4 3 5 8 31
|
brtxpsd3 |
|- ( <. A , B >. Cup C <-> C = ( A u. B ) ) |