MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tskmid Structured version   Unicode version

Theorem tskmid 9207
Description: The set  A is an element of the smallest Tarski 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 22 . . . 4  |-  ( A  e.  x  ->  A  e.  x )
21rgenw 2815 . . 3  |-  A. x  e.  Tarski  ( A  e.  x  ->  A  e.  x )
3 elintrabg 4284 . . 3  |-  ( A  e.  V  ->  ( A  e.  |^| { x  e.  Tarski  |  A  e.  x }  <->  A. x  e.  Tarski  ( A  e.  x  ->  A  e.  x ) ) )
42, 3mpbiri 233 . 2  |-  ( A  e.  V  ->  A  e.  |^| { x  e. 
Tarski  |  A  e.  x } )
5 tskmval 9206 . 2  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
64, 5eleqtrrd 2545 1  |-  ( A  e.  V  ->  A  e.  ( tarskiMap `  A )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1823   A.wral 2804   {crab 2808   |^|cint 4271   ` cfv 5570   Tarskictsk 9115   tarskiMapctskm 9204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-groth 9190
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-tsk 9116  df-tskm 9205
This theorem is referenced by:  eltskm  9210
  Copyright terms: Public domain W3C validator