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Theorem 0sdomg 7165
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 7163 . . 3  |-  ( A  e.  V  ->  (/)  ~<_  A )
2 brsdom 7059 . . . 4  |-  ( (/)  ~<  A 
<->  ( (/)  ~<_  A  /\  -.  (/)  ~~  A )
)
32baib 872 . . 3  |-  ( (/)  ~<_  A  ->  ( (/)  ~<  A  <->  -.  (/)  ~~  A
) )
41, 3syl 16 . 2  |-  ( A  e.  V  ->  ( (/) 
~<  A  <->  -.  (/)  ~~  A
) )
5 ensymb 7084 . . . 4  |-  ( (/)  ~~  A  <->  A  ~~  (/) )
6 en0 7099 . . . 4  |-  ( A 
~~  (/)  <->  A  =  (/) )
75, 6bitri 241 . . 3  |-  ( (/)  ~~  A  <->  A  =  (/) )
87necon3bbii 2574 . 2  |-  ( -.  (/)  ~~  A  <->  A  =/=  (/) )
94, 8syl6bb 253 1  |-  ( A  e.  V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717    =/= wne 2543   (/)c0 3564   class class class wbr 4146    ~~ cen 7035    ~<_ cdom 7036    ~< csdm 7037
This theorem is referenced by:  0sdom  7167  fodomr  7187  pwdom  7188  sdom1  7237  infn0  7298  fodomfib  7315  domwdom  7468  iunfictbso  7921  cdalepw  8002  fin45  8198  fodomb  8330  brdom3  8332  gchxpidm  8470  inar1  8576  csdfil  17840  ovoliunnul  19263  ovoliunnfl  25946  voliunnfl  25948  volsupnfl  25949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041
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