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Theorem tskin 10776
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 4224 . 2 (𝐴𝐵) ⊆ 𝐴
2 tskss 10775 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇 ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐴𝐵) ∈ 𝑇)
31, 2mp3an3 1447 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐴𝐵) ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  cin 3944  wss 3945  Tarskictsk 10765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-tsk 10766
This theorem is referenced by: (None)
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