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Theorem tsk0 8639
Description: A non-empty Tarski's class contains the empty set. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 18-Jun-2013.)
Assertion
Ref Expression
tsk0  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (/)  e.  T
)

Proof of Theorem tsk0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 n0 3638 . . 3  |-  ( T  =/=  (/)  <->  E. x  x  e.  T )
2 0ss 3657 . . . . . 6  |-  (/)  C_  x
3 tskss 8634 . . . . . 6  |-  ( ( T  e.  Tarski  /\  x  e.  T  /\  (/)  C_  x
)  ->  (/)  e.  T
)
42, 3mp3an3 1269 . . . . 5  |-  ( ( T  e.  Tarski  /\  x  e.  T )  ->  (/)  e.  T
)
54expcom 426 . . . 4  |-  ( x  e.  T  ->  ( T  e.  Tarski  ->  (/)  e.  T
) )
65exlimiv 1645 . . 3  |-  ( E. x  x  e.  T  ->  ( T  e.  Tarski  ->  (/) 
e.  T ) )
71, 6sylbi 189 . 2  |-  ( T  =/=  (/)  ->  ( T  e.  Tarski  ->  (/)  e.  T ) )
87impcom 421 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (/)  e.  T
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551    e. wcel 1726    =/= wne 2600    C_ wss 3321   (/)c0 3629   Tarskictsk 8624
This theorem is referenced by:  tsk1  8640  tskr1om  8643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-tsk 8625
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