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Theorem tsktrss 9789
 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 1131 . . 3 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → Tr 𝐴)
2 dftr4 4892 . . 3 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
31, 2sylib 208 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝐴 ⊆ 𝒫 𝐴)
4 tskpwss 9780 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
543adant2 1125 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝒫 𝐴𝑇)
63, 5sstrd 3762 1 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝐴𝑇)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1071   ∈ wcel 2145   ⊆ wss 3723  𝒫 cpw 4298  Tr wtr 4887  Tarskictsk 9776 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-tr 4888  df-tsk 9777 This theorem is referenced by: (None)
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