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Theorem subtareqbe 25994
Description: If  A is a subset of the smallest Tarski's class that contains  X then it is equipotent to this class or it belongs to it. CLASSES1 th. 8. (Contributed by FL, 17-Apr-2011.)
Assertion
Ref Expression
subtareqbe  |-  ( A 
C_  ( tarskiMap `  X
)  ->  ( A  ~~  ( tarskiMap `  X )  \/  A  e.  ( tarskiMap `  X ) ) )

Proof of Theorem subtareqbe
StepHypRef Expression
1 tskmcl 8479 . 2  |-  ( tarskiMap `  X )  e.  Tarski
2 tsken 8392 . 2  |-  ( ( ( tarskiMap `  X )  e.  Tarski  /\  A  C_  ( tarskiMap `  X ) )  -> 
( A  ~~  ( tarskiMap `  X )  \/  A  e.  ( tarskiMap `  X )
) )
31, 2mpan 651 1  |-  ( A 
C_  ( tarskiMap `  X
)  ->  ( A  ~~  ( tarskiMap `  X )  \/  A  e.  ( tarskiMap `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    e. wcel 1696    C_ wss 3165   class class class wbr 4039   ` cfv 5271    ~~ cen 6876   Tarskictsk 8386   tarskiMapctskm 8475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-groth 8461
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-tsk 8387  df-tskm 8476
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