Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep , we invoke the Axiom of Extensionality (indirectly via vtocl ), which is needed for the justification of class variable notation.
If we omit the requirement that y not occur in ph , we can derive a contradiction, as notzfaus shows. (Contributed by NM, 21-Jun-1993)
Ref | Expression | ||
---|---|---|---|
Hypothesis | zfauscl.1 | |- A e. _V |
|
Assertion | zfauscl | |- E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfauscl.1 | |- A e. _V |
|
2 | eleq2 | |- ( z = A -> ( x e. z <-> x e. A ) ) |
|
3 | 2 | anbi1d | |- ( z = A -> ( ( x e. z /\ ph ) <-> ( x e. A /\ ph ) ) ) |
4 | 3 | bibi2d | |- ( z = A -> ( ( x e. y <-> ( x e. z /\ ph ) ) <-> ( x e. y <-> ( x e. A /\ ph ) ) ) ) |
5 | 4 | albidv | |- ( z = A -> ( A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> A. x ( x e. y <-> ( x e. A /\ ph ) ) ) ) |
6 | 5 | exbidv | |- ( z = A -> ( E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) ) ) |
7 | ax-sep | |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) |
|
8 | 1 6 7 | vtocl | |- E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) |