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Theorem dom0 7238
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 7118 . . . . 5  |-  Rel  ~<_
21brrelexi 4921 . . . 4  |-  ( A  ~<_  (/)  ->  A  e.  _V )
3 0domg 7237 . . . 4  |-  ( A  e.  _V  ->  (/)  ~<_  A )
42, 3syl 16 . . 3  |-  ( A  ~<_  (/)  ->  (/)  ~<_  A )
54pm4.71i 615 . 2  |-  ( A  ~<_  (/) 
<->  ( A  ~<_  (/)  /\  (/)  ~<_  A ) )
6 sbthb 7231 . 2  |-  ( ( A  ~<_  (/)  /\  (/)  ~<_  A )  <-> 
A  ~~  (/) )
7 en0 7173 . 2  |-  ( A 
~~  (/)  <->  A  =  (/) )
85, 6, 73bitri 264 1  |-  ( A  ~<_  (/) 
<->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   class class class wbr 4215    ~~ cen 7109    ~<_ cdom 7110
This theorem is referenced by:  pwcdadom  8101  fin1a2lem11  8295  cfpwsdom  8464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-er 6908  df-en 7113  df-dom 7114
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