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

Theorem tskssel 8424
Description: A part of a Tarski's 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  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  A  e.  T )

Proof of Theorem tskssel
StepHypRef Expression
1 sdomnen 6933 . . 3  |-  ( A 
~<  T  ->  -.  A  ~~  T )
213ad2ant3 978 . 2  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  -.  A  ~~  T )
3 tsken 8421 . . . 4  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  ( A  ~~  T  \/  A  e.  T ) )
433adant3 975 . . 3  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  ( A  ~~  T  \/  A  e.  T ) )
54ord 366 . 2  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  ( -.  A  ~~  T  ->  A  e.  T ) )
62, 5mpd 14 1  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  A  ~<  T )  ->  A  e.  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ w3a 934    e. wcel 1701    C_ wss 3186   class class class wbr 4060    ~~ cen 6903    ~< csdm 6905   Tarskictsk 8415
This theorem is referenced by:  tskpr  8437  tskwe2  8440  tskord  8447  tskcard  8448  tskurn  8456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-sdom 6909  df-tsk 8416
  Copyright terms: Public domain W3C validator