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Theorem dom0 7725
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 7600 . . . . 5  |-  Rel  ~<_
21brrelexi 4893 . . . 4  |-  ( A  ~<_  (/)  ->  A  e.  _V )
3 0domg 7724 . . . 4  |-  ( A  e.  _V  ->  (/)  ~<_  A )
42, 3syl 17 . . 3  |-  ( A  ~<_  (/)  ->  (/)  ~<_  A )
54pm4.71i 642 . 2  |-  ( A  ~<_  (/) 
<->  ( A  ~<_  (/)  /\  (/)  ~<_  A ) )
6 sbthb 7718 . 2  |-  ( ( A  ~<_  (/)  /\  (/)  ~<_  A )  <-> 
A  ~~  (/) )
7 en0 7657 . 2  |-  ( A 
~~  (/)  <->  A  =  (/) )
85, 6, 73bitri 279 1  |-  ( A  ~<_  (/) 
<->  A  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897   _Vcvv 3056   (/)c0 3742   class class class wbr 4415    ~~ cen 7591    ~<_ cdom 7592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-er 7388  df-en 7595  df-dom 7596
This theorem is referenced by:  pwcdadom  8671  fin1a2lem11  8865  cfpwsdom  9034
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