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Theorem 0sdomg 7647
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 7645 . . 3  |-  ( A  e.  V  ->  (/)  ~<_  A )
2 brsdom 7539 . . . 4  |-  ( (/)  ~<  A 
<->  ( (/)  ~<_  A  /\  -.  (/)  ~~  A )
)
32baib 901 . . 3  |-  ( (/)  ~<_  A  ->  ( (/)  ~<  A  <->  -.  (/)  ~~  A
) )
41, 3syl 16 . 2  |-  ( A  e.  V  ->  ( (/) 
~<  A  <->  -.  (/)  ~~  A
) )
5 ensymb 7564 . . . 4  |-  ( (/)  ~~  A  <->  A  ~~  (/) )
6 en0 7579 . . . 4  |-  ( A 
~~  (/)  <->  A  =  (/) )
75, 6bitri 249 . . 3  |-  ( (/)  ~~  A  <->  A  =  (/) )
87necon3bbii 2728 . 2  |-  ( -.  (/)  ~~  A  <->  A  =/=  (/) )
94, 8syl6bb 261 1  |-  ( A  e.  V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    =/= wne 2662   (/)c0 3785   class class class wbr 4447    ~~ cen 7514    ~<_ cdom 7515    ~< csdm 7516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520
This theorem is referenced by:  0sdom  7649  fodomr  7669  pwdom  7670  sdom1  7720  infn0  7783  fodomfib  7801  domwdom  8001  iunfictbso  8496  cdalepw  8577  fin45  8773  fodomb  8905  brdom3  8907  gchxpidm  9048  inar1  9154  csdfil  20222  ovoliunnul  21745  ovoliunnfl  29909  voliunnfl  29911  volsupnfl  29912
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