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Theorem 0wdom 7284
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 2283 . . 3  |-  (/)  =  (/)
21orci 379 . 2  |-  ( (/)  =  (/)  \/  E. z 
z : X -onto-> (/) )
3 brwdom 7281 . 2  |-  ( X  e.  V  ->  ( (/)  ~<_*  X 
<->  ( (/)  =  (/)  \/  E. z  z : X -onto-> (/) ) ) )
42, 3mpbiri 224 1  |-  ( X  e.  V  ->  (/)  ~<_*  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357   E.wex 1528    = wceq 1623    e. wcel 1684   (/)c0 3455   class class class wbr 4023   -onto->wfo 5253    ~<_* cwdom 7271
This theorem is referenced by:  brwdom2  7287  wdomtr  7289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-fn 5258  df-fo 5261  df-wdom 7273
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