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Theorem tskin 10696
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 4189 . 2 (𝐴𝐵) ⊆ 𝐴
2 tskss 10695 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇 ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴𝐵) ∈ 𝑇)
31, 2mp3an3 1451 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐴𝐵) ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  cin 3910  wss 3911  Tarskictsk 10685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-tsk 10686
This theorem is referenced by: (None)
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