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Theorem tskmid 8462
Description: The set  A is an element of the smallest Tarski's class that contains  A. CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
Assertion
Ref Expression
tskmid  |-  ( A  e.  V  ->  A  e.  ( tarskiMap `  A )
)

Proof of Theorem tskmid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 19 . . . 4  |-  ( A  e.  x  ->  A  e.  x )
21rgenw 2610 . . 3  |-  A. x  e.  Tarski  ( A  e.  x  ->  A  e.  x )
3 elintrabg 3875 . . 3  |-  ( A  e.  V  ->  ( A  e.  |^| { x  e.  Tarski  |  A  e.  x }  <->  A. x  e.  Tarski  ( A  e.  x  ->  A  e.  x ) ) )
42, 3mpbiri 224 . 2  |-  ( A  e.  V  ->  A  e.  |^| { x  e. 
Tarski  |  A  e.  x } )
5 tskmval 8461 . 2  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
64, 5eleqtrrd 2360 1  |-  ( A  e.  V  ->  A  e.  ( tarskiMap `  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   A.wral 2543   {crab 2547   |^|cint 3862   ` cfv 5255   Tarskictsk 8370   tarskiMapctskm 8459
This theorem is referenced by:  eltskm  8465  tartarmap  25888
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-groth 8445
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-eu 2147  df-mo 2148  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-sbc 2992  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-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-tsk 8371  df-tskm 8460
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