Step |
Hyp |
Ref |
Expression |
1 |
|
indif1 |
|- ( ( A \ C ) i^i B ) = ( ( A i^i B ) \ C ) |
2 |
1
|
eqeq1i |
|- ( ( ( A \ C ) i^i B ) = (/) <-> ( ( A i^i B ) \ C ) = (/) ) |
3 |
|
ssdif0 |
|- ( ( A i^i B ) C_ C <-> ( ( A i^i B ) \ C ) = (/) ) |
4 |
2 3
|
sylbb2 |
|- ( ( ( A \ C ) i^i B ) = (/) -> ( A i^i B ) C_ C ) |
5 |
4
|
adantr |
|- ( ( ( ( A \ C ) i^i B ) = (/) /\ ( ( C \ A ) i^i B ) = (/) ) -> ( A i^i B ) C_ C ) |
6 |
|
inss2 |
|- ( A i^i B ) C_ B |
7 |
6
|
a1i |
|- ( ( ( ( A \ C ) i^i B ) = (/) /\ ( ( C \ A ) i^i B ) = (/) ) -> ( A i^i B ) C_ B ) |
8 |
5 7
|
ssind |
|- ( ( ( ( A \ C ) i^i B ) = (/) /\ ( ( C \ A ) i^i B ) = (/) ) -> ( A i^i B ) C_ ( C i^i B ) ) |
9 |
|
indif1 |
|- ( ( C \ A ) i^i B ) = ( ( C i^i B ) \ A ) |
10 |
9
|
eqeq1i |
|- ( ( ( C \ A ) i^i B ) = (/) <-> ( ( C i^i B ) \ A ) = (/) ) |
11 |
|
ssdif0 |
|- ( ( C i^i B ) C_ A <-> ( ( C i^i B ) \ A ) = (/) ) |
12 |
10 11
|
sylbb2 |
|- ( ( ( C \ A ) i^i B ) = (/) -> ( C i^i B ) C_ A ) |
13 |
12
|
adantl |
|- ( ( ( ( A \ C ) i^i B ) = (/) /\ ( ( C \ A ) i^i B ) = (/) ) -> ( C i^i B ) C_ A ) |
14 |
|
inss2 |
|- ( C i^i B ) C_ B |
15 |
14
|
a1i |
|- ( ( ( ( A \ C ) i^i B ) = (/) /\ ( ( C \ A ) i^i B ) = (/) ) -> ( C i^i B ) C_ B ) |
16 |
13 15
|
ssind |
|- ( ( ( ( A \ C ) i^i B ) = (/) /\ ( ( C \ A ) i^i B ) = (/) ) -> ( C i^i B ) C_ ( A i^i B ) ) |
17 |
8 16
|
eqssd |
|- ( ( ( ( A \ C ) i^i B ) = (/) /\ ( ( C \ A ) i^i B ) = (/) ) -> ( A i^i B ) = ( C i^i B ) ) |