Description: A subclass of a set is a set. Exercise 3 of TakeutiZaring p. 22 (generalized). (Contributed by NM, 14-Aug-1994) (Proof shortened by BJ, 18-Jul-2026)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ssexg | |- ( ( A C_ B /\ B e. C ) -> A e. _V ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfss2 | |- ( A C_ B <-> ( A i^i B ) = A ) |
|
| 2 | inex2g | |- ( B e. C -> ( A i^i B ) e. _V ) |
|
| 3 | eleq1 | |- ( ( A i^i B ) = A -> ( ( A i^i B ) e. _V <-> A e. _V ) ) |
|
| 4 | 3 | biimpa | |- ( ( ( A i^i B ) = A /\ ( A i^i B ) e. _V ) -> A e. _V ) |
| 5 | 2 4 | sylan2 | |- ( ( ( A i^i B ) = A /\ B e. C ) -> A e. _V ) |
| 6 | 1 5 | sylanb | |- ( ( A C_ B /\ B e. C ) -> A e. _V ) |