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Theorem tskss 10372
Description: The subsets of an element of a Tarski class belong to the class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 18-Jun-2013.)
Assertion
Ref Expression
tskss ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐵𝐴) → 𝐵𝑇)

Proof of Theorem tskss
StepHypRef Expression
1 elpw2g 5237 . . . 4 (𝐴𝑇 → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
21adantl 485 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
3 tskpwss 10366 . . . 4 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
43sseld 3900 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐵 ∈ 𝒫 𝐴𝐵𝑇))
52, 4sylbird 263 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐵𝐴𝐵𝑇))
653impia 1119 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐵𝐴) → 𝐵𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089  wcel 2110  wss 3866  𝒫 cpw 4513  Tarskictsk 10362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-tsk 10363
This theorem is referenced by:  tskin  10373  tsksn  10374  tsksuc  10376  tsk0  10377  tskr1om2  10382  tskint  10399
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