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Theorem tskin 8426
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 3423 . 2  |-  ( A  i^i  B )  C_  A
2 tskss 8425 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  ( A  i^i  B )  C_  A )  ->  ( A  i^i  B )  e.  T )
31, 2mp3an3 1266 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( A  i^i  B )  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1701    i^i cin 3185    C_ wss 3186   Tarskictsk 8415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-tsk 8416
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