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Theorem tskssel 8632
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 7136 . . 3  |-  ( A 
~<  T  ->  -.  A  ~~  T )
213ad2ant3 980 . 2  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  -.  A  ~~  T )
3 tsken 8629 . . . 4  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  ( A  ~~  T  \/  A  e.  T ) )
433adant3 977 . . 3  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  ( A  ~~  T  \/  A  e.  T ) )
54ord 367 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ w3a 936    e. wcel 1725    C_ wss 3320   class class class wbr 4212    ~~ cen 7106    ~< csdm 7108   Tarskictsk 8623
This theorem is referenced by:  tskpr  8645  tskwe2  8648  tskord  8655  tskcard  8656  tskurn  8664
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-sdom 7112  df-tsk 8624
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