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Theorem 0wdom 7999
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 2443 . . 3  |-  (/)  =  (/)
21orci 390 . 2  |-  ( (/)  =  (/)  \/  E. z 
z : X -onto-> (/) )
3 brwdom 7996 . 2  |-  ( X  e.  V  ->  ( (/)  ~<_*  X 
<->  ( (/)  =  (/)  \/  E. z  z : X -onto-> (/) ) ) )
42, 3mpbiri 233 1  |-  ( X  e.  V  ->  (/)  ~<_*  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1383   E.wex 1599    e. wcel 1804   (/)c0 3770   class class class wbr 4437   -onto->wfo 5576    ~<_* cwdom 7986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-cnv 4997  df-dm 4999  df-rn 5000  df-fn 5581  df-fo 5584  df-wdom 7988
This theorem is referenced by:  brwdom2  8002  wdomtr  8004
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