Description: Two ways to say that a collection B ( i ) for i e. A is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016)
Ref | Expression | ||
---|---|---|---|
Assertion | disjors | |- ( Disj_ x e. A B <-> A. i e. A A. j e. A ( i = j \/ ( [_ i / x ]_ B i^i [_ j / x ]_ B ) = (/) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv | |- F/_ i B |
|
2 | nfcsb1v | |- F/_ x [_ i / x ]_ B |
|
3 | csbeq1a | |- ( x = i -> B = [_ i / x ]_ B ) |
|
4 | 1 2 3 | cbvdisj | |- ( Disj_ x e. A B <-> Disj_ i e. A [_ i / x ]_ B ) |
5 | csbeq1 | |- ( i = j -> [_ i / x ]_ B = [_ j / x ]_ B ) |
|
6 | 5 | disjor | |- ( Disj_ i e. A [_ i / x ]_ B <-> A. i e. A A. j e. A ( i = j \/ ( [_ i / x ]_ B i^i [_ j / x ]_ B ) = (/) ) ) |
7 | 4 6 | bitri | |- ( Disj_ x e. A B <-> A. i e. A A. j e. A ( i = j \/ ( [_ i / x ]_ B i^i [_ j / x ]_ B ) = (/) ) ) |