MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0sdomg Unicode version

Theorem 0sdomg 7006
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 7004 . . 3  |-  ( A  e.  V  ->  (/)  ~<_  A )
2 brsdom 6900 . . . 4  |-  ( (/)  ~<  A 
<->  ( (/)  ~<_  A  /\  -.  (/)  ~~  A )
)
32baib 871 . . 3  |-  ( (/)  ~<_  A  ->  ( (/)  ~<  A  <->  -.  (/)  ~~  A
) )
41, 3syl 15 . 2  |-  ( A  e.  V  ->  ( (/) 
~<  A  <->  -.  (/)  ~~  A
) )
5 ensymb 6925 . . . 4  |-  ( (/)  ~~  A  <->  A  ~~  (/) )
6 en0 6940 . . . 4  |-  ( A 
~~  (/)  <->  A  =  (/) )
75, 6bitri 240 . . 3  |-  ( (/)  ~~  A  <->  A  =  (/) )
87necon3bbii 2490 . 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 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   class class class wbr 4039    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878
This theorem is referenced by:  0sdom  7008  fodomr  7028  pwdom  7029  sdom1  7078  infn0  7135  fodomfib  7152  domwdom  7304  iunfictbso  7757  cdalepw  7838  fin45  8034  fodomb  8167  brdom3  8169  gchxpidm  8307  inar1  8413  csdfil  17605  ovoliunnul  18882  snct  23354  ovoliunnfl  25001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882
  Copyright terms: Public domain W3C validator