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Theorem tsktrss 10618
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 1136 . . 3 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → Tr 𝐴)
2 dftr4 5216 . . 3 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
31, 2sylib 217 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝐴 ⊆ 𝒫 𝐴)
4 tskpwss 10609 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
543adant2 1130 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝒫 𝐴𝑇)
63, 5sstrd 3942 1 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086  wcel 2105  wss 3898  𝒫 cpw 4547  Tr wtr 5209  Tarskictsk 10605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-tr 5210  df-tsk 10606
This theorem is referenced by: (None)
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