Description: The subset of a set is also a set. Exercise 3 of TakeutiZaring p. 22. This is one way to express the Axiom of Separation ax-sep (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | ssex.1 | |- B e. _V |
|
| Assertion | ssex | |- ( A C_ B -> A e. _V ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssex.1 | |- B e. _V |
|
| 2 | df-ss | |- ( A C_ B <-> ( A i^i B ) = A ) |
|
| 3 | 1 | inex2 | |- ( A i^i B ) e. _V |
| 4 | eleq1 | |- ( ( A i^i B ) = A -> ( ( A i^i B ) e. _V <-> A e. _V ) ) |
|
| 5 | 3 4 | mpbii | |- ( ( A i^i B ) = A -> A e. _V ) |
| 6 | 2 5 | sylbi | |- ( A C_ B -> A e. _V ) |