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Theorem tsksn 8382
Description: A singleton of an element of a Tarski's class belongs to the class. JFM CLASSES2 th. 2 (partly) (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 18-Jun-2013.)
Assertion
Ref Expression
tsksn  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  { A }  e.  T )

Proof of Theorem tsksn
StepHypRef Expression
1 tskpw 8375 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  e.  T )
2 snsspw 3784 . . 3  |-  { A }  C_  ~P A
3 tskss 8380 . . 3  |-  ( ( T  e.  Tarski  /\  ~P A  e.  T  /\  { A }  C_  ~P A )  ->  { A }  e.  T )
42, 3mp3an3 1266 . 2  |-  ( ( T  e.  Tarski  /\  ~P A  e.  T )  ->  { A }  e.  T )
51, 4syldan 456 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  { A }  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   {csn 3640   Tarskictsk 8370
This theorem is referenced by:  tsk1  8386  tskop  8393
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-pow 4188
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-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
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