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Theorem tsksdom 8595
Description: An 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 7228 . . 3  |-  ( A  e.  T  ->  A  ~<  ~P A )
21adantl 453 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  ~P A )
3 simpl 444 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  T  e.  Tarski )
4 tskpwss 8591 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )
5 ssdomg 7120 . . 3  |-  ( T  e.  Tarski  ->  ( ~P A  C_  T  ->  ~P A  ~<_  T ) )
63, 4, 5sylc 58 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  ~<_  T )
7 sdomdomtr 7207 . 2  |-  ( ( A  ~<  ~P A  /\  ~P A  ~<_  T )  ->  A  ~<  T )
82, 6, 7syl2anc 643 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721    C_ wss 3288   ~Pcpw 3767   class class class wbr 4180    ~<_ cdom 7074    ~< csdm 7075   Tarskictsk 8587
This theorem is referenced by:  2domtsk  8605  r1tskina  8621  tskuni  8622  tskurn  8628  inaprc  8675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-tsk 8588
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