Description: The predicate "is an atom". ( ela analog.) (Contributed by NM, 18-Sep-2011)
Ref | Expression | ||
---|---|---|---|
Hypotheses | isatom.b | |- B = ( Base ` K ) |
|
isatom.z | |- .0. = ( 0. ` K ) |
||
isatom.c | |- C = ( |
||
isatom.a | |- A = ( Atoms ` K ) |
||
Assertion | isat | |- ( K e. D -> ( P e. A <-> ( P e. B /\ .0. C P ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isatom.b | |- B = ( Base ` K ) |
|
2 | isatom.z | |- .0. = ( 0. ` K ) |
|
3 | isatom.c | |- C = ( |
|
4 | isatom.a | |- A = ( Atoms ` K ) |
|
5 | 1 2 3 4 | pats | |- ( K e. D -> A = { x e. B | .0. C x } ) |
6 | 5 | eleq2d | |- ( K e. D -> ( P e. A <-> P e. { x e. B | .0. C x } ) ) |
7 | breq2 | |- ( x = P -> ( .0. C x <-> .0. C P ) ) |
|
8 | 7 | elrab | |- ( P e. { x e. B | .0. C x } <-> ( P e. B /\ .0. C P ) ) |
9 | 6 8 | bitrdi | |- ( K e. D -> ( P e. A <-> ( P e. B /\ .0. C P ) ) ) |