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Theorem dmsnn0 5479
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 3121 . . . . 5  |-  x  e. 
_V
21eldm 5206 . . . 4  |-  ( x  e.  dom  { A } 
<->  E. y  x { A } y )
3 df-br 4454 . . . . . 6  |-  ( x { A } y  <->  <. x ,  y >.  e.  { A } )
4 opex 4717 . . . . . . 7  |-  <. x ,  y >.  e.  _V
54elsnc 4057 . . . . . 6  |-  ( <.
x ,  y >.  e.  { A }  <->  <. x ,  y >.  =  A
)
6 eqcom 2476 . . . . . 6  |-  ( <.
x ,  y >.  =  A  <->  A  =  <. x ,  y >. )
73, 5, 63bitri 271 . . . . 5  |-  ( x { A } y  <-> 
A  =  <. x ,  y >. )
87exbii 1644 . . . 4  |-  ( E. y  x { A } y  <->  E. y  A  =  <. x ,  y >. )
92, 8bitr2i 250 . . 3  |-  ( E. y  A  =  <. x ,  y >.  <->  x  e.  dom  { A } )
109exbii 1644 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>. 
<->  E. x  x  e. 
dom  { A } )
11 elvv 5064 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
12 n0 3799 . 2  |-  ( dom 
{ A }  =/=  (/)  <->  E. x  x  e.  dom  { A } )
1310, 11, 123bitr4i 277 1  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   _Vcvv 3118   (/)c0 3790   {csn 4033   <.cop 4039   class class class wbr 4453    X. cxp 5003   dom cdm 5005
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
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-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-dm 5015
This theorem is referenced by:  rnsnn0  5480  dmsn0  5481  dmsn0el  5483  relsn2  5484  1stnpr  6799  1st2val  6821  mpt2xopxnop0  6955  hashfun  12476
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