MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfdom2 Unicode version

Theorem dfdom2 6889
Description: Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)
Assertion
Ref Expression
dfdom2  |-  ~<_  =  ( 
~<  u.  ~~  )

Proof of Theorem dfdom2
StepHypRef Expression
1 df-sdom 6868 . . 3  |-  ~<  =  (  ~<_  \  ~~  )
21uneq2i 3328 . 2  |-  (  ~~  u.  ~<  )  =  ( 
~~  u.  (  ~<_  \  ~~  ) )
3 uncom 3321 . 2  |-  (  ~~  u.  ~<  )  =  ( 
~<  u.  ~~  )
4 enssdom 6888 . . 3  |-  ~~  C_  ~<_
5 undif 3536 . . 3  |-  (  ~~  C_  ~<_  <-> 
(  ~~  u.  (  ~<_  \ 
~~  ) )  =  ~<_  )
64, 5mpbi 199 . 2  |-  (  ~~  u.  (  ~<_  \  ~~  )
)  =  ~<_
72, 3, 63eqtr3ri 2314 1  |-  ~<_  =  ( 
~<  u.  ~~  )
Colors of variables: wff set class
Syntax hints:    = wceq 1625    \ cdif 3151    u. cun 3152    C_ wss 3154    ~~ cen 6862    ~<_ cdom 6863    ~< csdm 6864
This theorem is referenced by:  brdom2  6893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-opab 4080  df-xp 4697  df-rel 4698  df-f1o 5264  df-en 6866  df-dom 6867  df-sdom 6868
  Copyright terms: Public domain W3C validator