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Theorem tsktrss 10183
Description: A transitive element of a Tarski class is a part of the class. JFM CLASSES2 th. 8. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tsktrss ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝐴𝑇)

Proof of Theorem tsktrss
StepHypRef Expression
1 simp2 1134 . . 3 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → Tr 𝐴)
2 dftr4 5164 . . 3 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
31, 2sylib 221 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝐴 ⊆ 𝒫 𝐴)
4 tskpwss 10174 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
543adant2 1128 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝒫 𝐴𝑇)
63, 5sstrd 3963 1 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2115  wss 3919  𝒫 cpw 4522  Tr wtr 5159  Tarskictsk 10170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-tr 5160  df-tsk 10171
This theorem is referenced by: (None)
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