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

Theorem tskssel 10167
Description: A part of a Tarski class strictly dominated by the class is an element of the class. JFM CLASSES2 th. 2. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tskssel ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → 𝐴𝑇)

Proof of Theorem tskssel
StepHypRef Expression
1 sdomnen 8526 . . 3 (𝐴𝑇 → ¬ 𝐴𝑇)
213ad2ant3 1127 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → ¬ 𝐴𝑇)
3 tsken 10164 . . . 4 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐴𝑇𝐴𝑇))
433adant3 1124 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → (𝐴𝑇𝐴𝑇))
54ord 858 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → (¬ 𝐴𝑇𝐴𝑇))
62, 5mpd 15 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 841  w3a 1079  wcel 2105  wss 3933   class class class wbr 5057  cen 8494  csdm 8496  Tarskictsk 10158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-sdom 8500  df-tsk 10159
This theorem is referenced by:  tskpr  10180  tskwe2  10183  tskord  10190  tskcard  10191  tskurn  10199
  Copyright terms: Public domain W3C validator