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Theorem 0sdom 7721
Description: A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 29-Jul-2004.)
Hypothesis
Ref Expression
0sdom.1  |-  A  e. 
_V
Assertion
Ref Expression
0sdom  |-  ( (/)  ~<  A 
<->  A  =/=  (/) )

Proof of Theorem 0sdom
StepHypRef Expression
1 0sdom.1 . 2  |-  A  e. 
_V
2 0sdomg 7719 . 2  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
31, 2ax-mp 5 1  |-  ( (/)  ~<  A 
<->  A  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    e. wcel 1904    =/= wne 2641   _Vcvv 3031   (/)c0 3722   class class class wbr 4395    ~< csdm 7586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590
This theorem is referenced by:  sdom1  7790  marypha1lem  7965  konigthlem  9011  pwcfsdom  9026  cfpwsdom  9027  rankcf  9220  r1tskina  9225  1stcfb  20537  snct  28370  sigapildsys  29058
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