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Theorem tsksdom 9180
Description: An element of a Tarski class is strictly dominated by the class. JFM CLASSES2 th. 1 (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 18-Jun-2013.)
Assertion
Ref Expression
tsksdom  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  T )

Proof of Theorem tsksdom
StepHypRef Expression
1 canth2g 7732 . . 3  |-  ( A  e.  T  ->  A  ~<  ~P A )
21adantl 467 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  ~P A )
3 simpl 458 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  T  e.  Tarski )
4 tskpwss 9176 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )
5 ssdomg 7622 . . 3  |-  ( T  e.  Tarski  ->  ( ~P A  C_  T  ->  ~P A  ~<_  T ) )
63, 4, 5sylc 62 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  ~<_  T )
7 sdomdomtr 7711 . 2  |-  ( ( A  ~<  ~P A  /\  ~P A  ~<_  T )  ->  A  ~<  T )
82, 6, 7syl2anc 665 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1870    C_ wss 3442   ~Pcpw 3985   class class class wbr 4426    ~<_ cdom 7575    ~< csdm 7576   Tarskictsk 9172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-tsk 9173
This theorem is referenced by:  2domtsk  9190  r1tskina  9206  tskuni  9207  tskurn  9213  inaprc  9260
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