Description: Any nonempty transitive class includes its intersection. Exercise 3 in TakeutiZaring p. 44 (which mistakenly does not include the nonemptiness hypothesis). (Contributed by Scott Fenton, 3-Mar-2011) (Proof shortened by Andrew Salmon, 14-Nov-2011)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | trintss | |- ( ( Tr A /\ A =/= (/) ) -> |^| A C_ A ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | n0 | |- ( A =/= (/) <-> E. x x e. A ) |
|
| 2 | intss1 | |- ( x e. A -> |^| A C_ x ) |
|
| 3 | trss | |- ( Tr A -> ( x e. A -> x C_ A ) ) |
|
| 4 | 3 | com12 | |- ( x e. A -> ( Tr A -> x C_ A ) ) |
| 5 | sstr2 | |- ( |^| A C_ x -> ( x C_ A -> |^| A C_ A ) ) |
|
| 6 | 2 4 5 | sylsyld | |- ( x e. A -> ( Tr A -> |^| A C_ A ) ) |
| 7 | 6 | exlimiv | |- ( E. x x e. A -> ( Tr A -> |^| A C_ A ) ) |
| 8 | 1 7 | sylbi | |- ( A =/= (/) -> ( Tr A -> |^| A C_ A ) ) |
| 9 | 8 | impcom | |- ( ( Tr A /\ A =/= (/) ) -> |^| A C_ A ) |