Description: If a class is a member of another class, then it is a set. Theorem 6.12 of Quine p. 44. (Contributed by NM, 26-May-1993) (Proof shortened by Andrew Salmon, 8-Jun-2011)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | elex | |- ( A e. B -> A e. _V ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | exsimpl | |- ( E. x ( x = A /\ x e. B ) -> E. x x = A ) |
|
| 2 | dfclel | |- ( A e. B <-> E. x ( x = A /\ x e. B ) ) |
|
| 3 | isset | |- ( A e. _V <-> E. x x = A ) |
|
| 4 | 1 2 3 | 3imtr4i | |- ( A e. B -> A e. _V ) |