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Theorem tskssel 9030
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 7443 . . 3  |-  ( A 
~<  T  ->  -.  A  ~~  T )
213ad2ant3 1011 . 2  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  -.  A  ~~  T )
3 tsken 9027 . . . 4  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  ( A  ~~  T  \/  A  e.  T ) )
433adant3 1008 . . 3  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  ( A  ~~  T  \/  A  e.  T ) )
54ord 377 . 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 368    /\ w3a 965    e. wcel 1758    C_ wss 3431   class class class wbr 4395    ~~ cen 7412    ~< csdm 7414   Tarskictsk 9021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-sdom 7418  df-tsk 9022
This theorem is referenced by:  tskpr  9043  tskwe2  9046  tskord  9053  tskcard  9054  tskurn  9062
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