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Theorem tskssel 9046
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.)
Assertion
Ref Expression
tskssel  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  A  e.  T )

Proof of Theorem tskssel
StepHypRef Expression
1 sdomnen 7463 . . 3  |-  ( A 
~<  T  ->  -.  A  ~~  T )
213ad2ant3 1017 . 2  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  -.  A  ~~  T )
3 tsken 9043 . . . 4  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  ( A  ~~  T  \/  A  e.  T ) )
433adant3 1014 . . 3  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  ( A  ~~  T  \/  A  e.  T ) )
54ord 375 . 2  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  ( -.  A  ~~  T  ->  A  e.  T ) )
62, 5mpd 15 1  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  A  e.  T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ w3a 971    e. wcel 1826    C_ wss 3389   class class class wbr 4367    ~~ cen 7432    ~< csdm 7434   Tarskictsk 9037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-sdom 7438  df-tsk 9038
This theorem is referenced by:  tskpr  9059  tskwe2  9062  tskord  9069  tskcard  9070  tskurn  9078
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