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Theorem dmsnn0 5292
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 2965 . . . . 5  |-  x  e. 
_V
21eldm 5024 . . . 4  |-  ( x  e.  dom  { A } 
<->  E. y  x { A } y )
3 df-br 4281 . . . . . 6  |-  ( x { A } y  <->  <. x ,  y >.  e.  { A } )
4 opex 4544 . . . . . . 7  |-  <. x ,  y >.  e.  _V
54elsnc 3889 . . . . . 6  |-  ( <.
x ,  y >.  e.  { A }  <->  <. x ,  y >.  =  A
)
6 eqcom 2435 . . . . . 6  |-  ( <.
x ,  y >.  =  A  <->  A  =  <. x ,  y >. )
73, 5, 63bitri 271 . . . . 5  |-  ( x { A } y  <-> 
A  =  <. x ,  y >. )
87exbii 1634 . . . 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 1634 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>. 
<->  E. x  x  e. 
dom  { A } )
11 elvv 4884 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
12 n0 3634 . 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 1362   E.wex 1589    e. wcel 1755    =/= wne 2596   _Vcvv 2962   (/)c0 3625   {csn 3865   <.cop 3871   class class class wbr 4280    X. cxp 4825   dom cdm 4827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-br 4281  df-opab 4339  df-xp 4833  df-dm 4837
This theorem is referenced by:  rnsnn0  5293  dmsn0  5294  dmsn0el  5296  relsn2  5297  1stnpr  6570  1st2val  6591  mpt2xopxnop0  6721  hashfun  12183
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