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Theorem 0sdomg 6990
Description: A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.)
Assertion
Ref Expression
0sdomg  |-  ( A  e.  V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )

Proof of Theorem 0sdomg
StepHypRef Expression
1 0domg 6988 . . 3  |-  ( A  e.  V  ->  (/)  ~<_  A )
2 brsdom 6884 . . . 4  |-  ( (/)  ~<  A 
<->  ( (/)  ~<_  A  /\  -.  (/)  ~~  A )
)
32baib 871 . . 3  |-  ( (/)  ~<_  A  ->  ( (/)  ~<  A  <->  -.  (/)  ~~  A
) )
41, 3syl 15 . 2  |-  ( A  e.  V  ->  ( (/) 
~<  A  <->  -.  (/)  ~~  A
) )
5 ensymb 6909 . . . 4  |-  ( (/)  ~~  A  <->  A  ~~  (/) )
6 en0 6924 . . . 4  |-  ( A 
~~  (/)  <->  A  =  (/) )
75, 6bitri 240 . . 3  |-  ( (/)  ~~  A  <->  A  =  (/) )
87necon3bbii 2477 . 2  |-  ( -.  (/)  ~~  A  <->  A  =/=  (/) )
94, 8syl6bb 252 1  |-  ( A  e.  V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   class class class wbr 4023    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862
This theorem is referenced by:  0sdom  6992  fodomr  7012  pwdom  7013  sdom1  7062  infn0  7119  fodomfib  7136  domwdom  7288  iunfictbso  7741  cdalepw  7822  fin45  8018  fodomb  8151  brdom3  8153  gchxpidm  8291  inar1  8397  csdfil  17589  ovoliunnul  18866  snct  23339
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  df-sdom 6866
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