MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0wdom Structured version   Unicode version

Theorem 0wdom 7540
Description: Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
Assertion
Ref Expression
0wdom  |-  ( X  e.  V  ->  (/)  ~<_*  X )

Proof of Theorem 0wdom
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . 3  |-  (/)  =  (/)
21orci 381 . 2  |-  ( (/)  =  (/)  \/  E. z 
z : X -onto-> (/) )
3 brwdom 7537 . 2  |-  ( X  e.  V  ->  ( (/)  ~<_*  X 
<->  ( (/)  =  (/)  \/  E. z  z : X -onto-> (/) ) ) )
42, 3mpbiri 226 1  |-  ( X  e.  V  ->  (/)  ~<_*  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359   E.wex 1551    = wceq 1653    e. wcel 1726   (/)c0 3630   class class class wbr 4214   -onto->wfo 5454    ~<_* cwdom 7527
This theorem is referenced by:  brwdom2  7543  wdomtr  7545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-dm 4890  df-rn 4891  df-fn 5459  df-fo 5462  df-wdom 7529
  Copyright terms: Public domain W3C validator