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Theorem dmsnn0 5138
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 2791 . . . . 5  |-  x  e. 
_V
21eldm 4876 . . . 4  |-  ( x  e.  dom  { A } 
<->  E. y  x { A } y )
3 df-br 4024 . . . . . 6  |-  ( x { A } y  <->  <. x ,  y >.  e.  { A } )
4 opex 4237 . . . . . . 7  |-  <. x ,  y >.  e.  _V
54elsnc 3663 . . . . . 6  |-  ( <.
x ,  y >.  e.  { A }  <->  <. x ,  y >.  =  A
)
6 eqcom 2285 . . . . . 6  |-  ( <.
x ,  y >.  =  A  <->  A  =  <. x ,  y >. )
73, 5, 63bitri 262 . . . . 5  |-  ( x { A } y  <-> 
A  =  <. x ,  y >. )
87exbii 1569 . . . 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 1569 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>. 
<->  E. x  x  e. 
dom  { A } )
11 elvv 4748 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
12 n0 3464 . 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 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   {csn 3640   <.cop 3643   class class class wbr 4023    X. cxp 4687   dom cdm 4689
This theorem is referenced by:  rnsnn0  5139  dmsn0  5140  dmsn0el  5142  relsn2  5143  1st2val  6145  hashfun  11389  1stnpr  23245  imfstnrelc  25081  mpt2xopxnop0  28081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699
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