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Theorem tsksn 8529
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 8522 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  e.  T )
2 snsspw 3884 . . 3  |-  { A }  C_  ~P A
3 tskss 8527 . . 3  |-  ( ( T  e.  Tarski  /\  ~P A  e.  T  /\  { A }  C_  ~P A )  ->  { A }  e.  T )
42, 3mp3an3 1267 . 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 1715    C_ wss 3238   ~Pcpw 3714   {csn 3729   Tarskictsk 8517
This theorem is referenced by:  tsk1  8533  tskop  8540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-pow 4290
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-tsk 8518
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