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Theorem dom0 6989
Description: A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
Assertion
Ref Expression
dom0  |-  ( A  ~<_  (/) 
<->  A  =  (/) )

Proof of Theorem dom0
StepHypRef Expression
1 reldom 6869 . . . . 5  |-  Rel  ~<_
21brrelexi 4729 . . . 4  |-  ( A  ~<_  (/)  ->  A  e.  _V )
3 0domg 6988 . . . 4  |-  ( A  e.  _V  ->  (/)  ~<_  A )
42, 3syl 15 . . 3  |-  ( A  ~<_  (/)  ->  (/)  ~<_  A )
54pm4.71i 613 . 2  |-  ( A  ~<_  (/) 
<->  ( A  ~<_  (/)  /\  (/)  ~<_  A ) )
6 sbthb 6982 . 2  |-  ( ( A  ~<_  (/)  /\  (/)  ~<_  A )  <-> 
A  ~~  (/) )
7 en0 6924 . 2  |-  ( A 
~~  (/)  <->  A  =  (/) )
85, 6, 73bitri 262 1  |-  ( A  ~<_  (/) 
<->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   class class class wbr 4023    ~~ cen 6860    ~<_ cdom 6861
This theorem is referenced by:  pwcdadom  7842  fin1a2lem11  8036  cfpwsdom  8206
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-pow 4188  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-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-er 6660  df-en 6864  df-dom 6865
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