Description: A part of a Tarski class strictly dominated by the class is an element of the class. JFM CLASSES2 th. 2. (Contributed by FL, 22-Feb-2011) (Proof shortened by Mario Carneiro, 20-Sep-2014)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | tskssel | |- ( ( T e. Tarski /\ A C_ T /\ A ~< T ) -> A e. T ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sdomnen | |- ( A ~< T -> -. A ~~ T ) |
|
| 2 | 1 | 3ad2ant3 | |- ( ( T e. Tarski /\ A C_ T /\ A ~< T ) -> -. A ~~ T ) |
| 3 | tsken | |- ( ( T e. Tarski /\ A C_ T ) -> ( A ~~ T \/ A e. T ) ) |
|
| 4 | 3 | 3adant3 | |- ( ( T e. Tarski /\ A C_ T /\ A ~< T ) -> ( A ~~ T \/ A e. T ) ) |
| 5 | 4 | ord | |- ( ( T e. Tarski /\ A C_ T /\ A ~< T ) -> ( -. A ~~ T -> A e. T ) ) |
| 6 | 2 5 | mpd | |- ( ( T e. Tarski /\ A C_ T /\ A ~< T ) -> A e. T ) |