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Theorem tskin 8947
Description: The intersection of two elements of a Tarski 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 3591 . 2  |-  ( A  i^i  B )  C_  A
2 tskss 8946 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  ( A  i^i  B )  C_  A )  ->  ( A  i^i  B )  e.  T )
31, 2mp3an3 1303 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( A  i^i  B )  e.  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756    i^i cin 3348    C_ wss 3349   Tarskictsk 8936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-br 4314  df-tsk 8937
This theorem is referenced by: (None)
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