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Theorem tsksdom 9033
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 7574 . . 3  |-  ( A  e.  T  ->  A  ~<  ~P A )
21adantl 466 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  ~P A )
3 simpl 457 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  T  e.  Tarski )
4 tskpwss 9029 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )
5 ssdomg 7464 . . 3  |-  ( T  e.  Tarski  ->  ( ~P A  C_  T  ->  ~P A  ~<_  T ) )
63, 4, 5sylc 60 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  ~<_  T )
7 sdomdomtr 7553 . 2  |-  ( ( A  ~<  ~P A  /\  ~P A  ~<_  T )  ->  A  ~<  T )
82, 6, 7syl2anc 661 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758    C_ wss 3435   ~Pcpw 3967   class class class wbr 4399    ~<_ cdom 7417    ~< csdm 7418   Tarskictsk 9025
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
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-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-tsk 9026
This theorem is referenced by:  2domtsk  9043  r1tskina  9059  tskuni  9060  tskurn  9066  inaprc  9113
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