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Theorem tskin 9154
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 3714 . 2  |-  ( A  i^i  B )  C_  A
2 tskss 9153 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  ( A  i^i  B )  C_  A )  ->  ( A  i^i  B )  e.  T )
31, 2mp3an3 1313 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 1819    i^i cin 3470    C_ wss 3471   Tarskictsk 9143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-tsk 9144
This theorem is referenced by: (None)
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