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

Theorem tsktrss 9528
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 1060 . . 3 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → Tr 𝐴)
2 dftr4 4722 . . 3 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
31, 2sylib 208 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝐴 ⊆ 𝒫 𝐴)
4 tskpwss 9519 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
543adant2 1078 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝒫 𝐴𝑇)
63, 5sstrd 3598 1 ((𝑇 ∈ Tarski ∧ Tr 𝐴𝐴𝑇) → 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036  wcel 1992  wss 3560  𝒫 cpw 4135  Tr wtr 4717  Tarskictsk 9515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-tr 4718  df-tsk 9516
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator