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Theorem 0sdomg 7239
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 7237 . . 3  |-  ( A  e.  V  ->  (/)  ~<_  A )
2 brsdom 7133 . . . 4  |-  ( (/)  ~<  A 
<->  ( (/)  ~<_  A  /\  -.  (/)  ~~  A )
)
32baib 873 . . 3  |-  ( (/)  ~<_  A  ->  ( (/)  ~<  A  <->  -.  (/)  ~~  A
) )
41, 3syl 16 . 2  |-  ( A  e.  V  ->  ( (/) 
~<  A  <->  -.  (/)  ~~  A
) )
5 ensymb 7158 . . . 4  |-  ( (/)  ~~  A  <->  A  ~~  (/) )
6 en0 7173 . . . 4  |-  ( A 
~~  (/)  <->  A  =  (/) )
75, 6bitri 242 . . 3  |-  ( (/)  ~~  A  <->  A  =  (/) )
87necon3bbii 2634 . 2  |-  ( -.  (/)  ~~  A  <->  A  =/=  (/) )
94, 8syl6bb 254 1  |-  ( A  e.  V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726    =/= wne 2601   (/)c0 3630   class class class wbr 4215    ~~ cen 7109    ~<_ cdom 7110    ~< csdm 7111
This theorem is referenced by:  0sdom  7241  fodomr  7261  pwdom  7262  sdom1  7311  infn0  7372  fodomfib  7389  domwdom  7545  iunfictbso  8000  cdalepw  8081  fin45  8277  fodomb  8409  brdom3  8411  gchxpidm  8549  inar1  8655  csdfil  17931  ovoliunnul  19408  ovoliunnfl  26260  voliunnfl  26262  volsupnfl  26263
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  df-sdom 7115
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