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Theorem tsksn 8599
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 8592 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  e.  T )
2 snsspw 3938 . . 3  |-  { A }  C_  ~P A
3 tskss 8597 . . 3  |-  ( ( T  e.  Tarski  /\  ~P A  e.  T  /\  { A }  C_  ~P A )  ->  { A }  e.  T )
42, 3mp3an3 1268 . 2  |-  ( ( T  e.  Tarski  /\  ~P A  e.  T )  ->  { A }  e.  T )
51, 4syldan 457 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  { A }  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721    C_ wss 3288   ~Pcpw 3767   {csn 3782   Tarskictsk 8587
This theorem is referenced by:  tsk1  8603  tskop  8610
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-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-pow 4345
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-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  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-br 4181  df-tsk 8588
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