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Theorem tskssel 8379
Description: A part of a Tarski's 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 6890 . . 3  |-  ( A 
~<  T  ->  -.  A  ~~  T )
213ad2ant3 978 . 2  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  -.  A  ~~  T )
3 tsken 8376 . . . 4  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  ( A  ~~  T  \/  A  e.  T ) )
433adant3 975 . . 3  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  ( A  ~~  T  \/  A  e.  T ) )
54ord 366 . 2  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  ( -.  A  ~~  T  ->  A  e.  T ) )
62, 5mpd 14 1  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  A  e.  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ w3a 934    e. wcel 1684    C_ wss 3152   class class class wbr 4023    ~~ cen 6860    ~< csdm 6862   Tarskictsk 8370
This theorem is referenced by:  tskpr  8392  tskwe2  8395  tskord  8402  tskcard  8403  tskurn  8411  fnctartar  25907  fnctartar2  25908  fnctartar3  25909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-sdom 6866  df-tsk 8371
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