Metamath Proof Explorer


Theorem tskssel

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 )

Proof

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 )