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Theorem tsksn 8666
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 8659 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  e.  T )
2 snsspw 3994 . . 3  |-  { A }  C_  ~P A
3 tskss 8664 . . 3  |-  ( ( T  e.  Tarski  /\  ~P A  e.  T  /\  { A }  C_  ~P A )  ->  { A }  e.  T )
42, 3mp3an3 1269 . 2  |-  ( ( T  e.  Tarski  /\  ~P A  e.  T )  ->  { A }  e.  T )
51, 4syldan 458 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  { A }  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1727    C_ wss 3306   ~Pcpw 3823   {csn 3838   Tarskictsk 8654
This theorem is referenced by:  tsk1  8670  tskop  8677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-pow 4406
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-tsk 8655
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