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Theorem tskin 8634
Description: The intersection of two elements of a Tarski's class belongs to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tskin  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( A  i^i  B )  e.  T )

Proof of Theorem tskin
StepHypRef Expression
1 inss1 3561 . 2  |-  ( A  i^i  B )  C_  A
2 tskss 8633 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  ( A  i^i  B )  C_  A )  ->  ( A  i^i  B )  e.  T )
31, 2mp3an3 1268 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( A  i^i  B )  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725    i^i cin 3319    C_ wss 3320   Tarskictsk 8623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-tsk 8624
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