MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tsktrss Structured version   Visualization version   GIF version

Theorem tsktrss 9462
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 1055 . . 3 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → Tr 𝐴)
2 dftr4 4685 . . 3 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
31, 2sylib 207 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝐴 ⊆ 𝒫 𝐴)
4 tskpwss 9453 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
543adant2 1073 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝒫 𝐴𝑇)
63, 5sstrd 3578 1 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1031  wcel 1977  wss 3540  𝒫 cpw 4108  Tr wtr 4680  Tarskictsk 9449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-tr 4681  df-tsk 9450
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator