MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tsk0 Unicode version

Theorem tsk0 8385
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 3464 . . 3  |-  ( T  =/=  (/)  <->  E. x  x  e.  T )
2 0ss 3483 . . . . . 6  |-  (/)  C_  x
3 tskss 8380 . . . . . 6  |-  ( ( T  e.  Tarski  /\  x  e.  T  /\  (/)  C_  x
)  ->  (/)  e.  T
)
42, 3mp3an3 1266 . . . . 5  |-  ( ( T  e.  Tarski  /\  x  e.  T )  ->  (/)  e.  T
)
54expcom 424 . . . 4  |-  ( x  e.  T  ->  ( T  e.  Tarski  ->  (/)  e.  T
) )
65exlimiv 1666 . . 3  |-  ( E. x  x  e.  T  ->  ( T  e.  Tarski  ->  (/) 
e.  T ) )
71, 6sylbi 187 . 2  |-  ( T  =/=  (/)  ->  ( T  e.  Tarski  ->  (/)  e.  T ) )
87impcom 419 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (/)  e.  T
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455   Tarskictsk 8370
This theorem is referenced by:  tsk1  8386  tskr1om  8389
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
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-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-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-br 4024  df-tsk 8371
  Copyright terms: Public domain W3C validator