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Theorem dmsnn0 5154
Description: The domain of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmsnn0  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )

Proof of Theorem dmsnn0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2804 . . . . 5  |-  x  e. 
_V
21eldm 4892 . . . 4  |-  ( x  e.  dom  { A } 
<->  E. y  x { A } y )
3 df-br 4040 . . . . . 6  |-  ( x { A } y  <->  <. x ,  y >.  e.  { A } )
4 opex 4253 . . . . . . 7  |-  <. x ,  y >.  e.  _V
54elsnc 3676 . . . . . 6  |-  ( <.
x ,  y >.  e.  { A }  <->  <. x ,  y >.  =  A
)
6 eqcom 2298 . . . . . 6  |-  ( <.
x ,  y >.  =  A  <->  A  =  <. x ,  y >. )
73, 5, 63bitri 262 . . . . 5  |-  ( x { A } y  <-> 
A  =  <. x ,  y >. )
87exbii 1572 . . . 4  |-  ( E. y  x { A } y  <->  E. y  A  =  <. x ,  y >. )
92, 8bitr2i 241 . . 3  |-  ( E. y  A  =  <. x ,  y >.  <->  x  e.  dom  { A } )
109exbii 1572 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>. 
<->  E. x  x  e. 
dom  { A } )
11 elvv 4764 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
12 n0 3477 . 2  |-  ( dom 
{ A }  =/=  (/)  <->  E. x  x  e.  dom  { A } )
1310, 11, 123bitr4i 268 1  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   (/)c0 3468   {csn 3653   <.cop 3656   class class class wbr 4039    X. cxp 4703   dom cdm 4705
This theorem is referenced by:  rnsnn0  5155  dmsn0  5156  dmsn0el  5158  relsn2  5159  1st2val  6161  hashfun  11405  1stnpr  23260  imfstnrelc  25184  mpt2xopxnop0  28197
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  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-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-dm 4715
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