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Theorem tsksdom 8378
Description: A element of a Tarski's 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 7015 . . 3  |-  ( A  e.  T  ->  A  ~<  ~P A )
21adantl 452 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  ~P A )
3 simpl 443 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  T  e.  Tarski )
4 tskpwss 8374 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )
5 ssdomg 6907 . . 3  |-  ( T  e.  Tarski  ->  ( ~P A  C_  T  ->  ~P A  ~<_  T ) )
63, 4, 5sylc 56 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  ~<_  T )
7 sdomdomtr 6994 . 2  |-  ( ( A  ~<  ~P A  /\  ~P A  ~<_  T )  ->  A  ~<  T )
82, 6, 7syl2anc 642 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023    ~<_ cdom 6861    ~< csdm 6862   Tarskictsk 8370
This theorem is referenced by:  2domtsk  8388  r1tskina  8404  tskuni  8405  tskurn  8411  inaprc  8458  carinttar  25902  cardtar  25904  fnctartar  25907  fnctartar2  25908
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-tsk 8371
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