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Theorem tskss 9792
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 4976 . . . 4 (𝐴𝑇 → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
21adantl 473 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
3 tskpwss 9786 . . . 4 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
43sseld 3743 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐵 ∈ 𝒫 𝐴𝐵𝑇))
52, 4sylbird 250 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐵𝐴𝐵𝑇))
653impia 1110 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐵𝐴) → 𝐵𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wcel 2139  wss 3715  𝒫 cpw 4302  Tarskictsk 9782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-tsk 9783
This theorem is referenced by:  tskin  9793  tsksn  9794  tsksuc  9796  tsk0  9797  tskr1om2  9802  tskint  9819
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