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Theorem tskin 10751
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 ∈ Tarski ∧ 𝐴𝑇) → (𝐴𝐵) ∈ 𝑇)

Proof of Theorem tskin
StepHypRef Expression
1 inss1 4221 . 2 (𝐴𝐵) ⊆ 𝐴
2 tskss 10750 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇 ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴𝐵) ∈ 𝑇)
31, 2mp3an3 1446 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐴𝐵) ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  cin 3940  wss 3941  Tarskictsk 10740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-tsk 10741
This theorem is referenced by: (None)
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