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Theorem dmsnn0 5415
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 3081 . . . . 5  |-  x  e. 
_V
21eldm 5148 . . . 4  |-  ( x  e.  dom  { A } 
<->  E. y  x { A } y )
3 df-br 4404 . . . . . 6  |-  ( x { A } y  <->  <. x ,  y >.  e.  { A } )
4 opex 4667 . . . . . . 7  |-  <. x ,  y >.  e.  _V
54elsnc 4012 . . . . . 6  |-  ( <.
x ,  y >.  e.  { A }  <->  <. x ,  y >.  =  A
)
6 eqcom 2463 . . . . . 6  |-  ( <.
x ,  y >.  =  A  <->  A  =  <. x ,  y >. )
73, 5, 63bitri 271 . . . . 5  |-  ( x { A } y  <-> 
A  =  <. x ,  y >. )
87exbii 1635 . . . 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 1635 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>. 
<->  E. x  x  e. 
dom  { A } )
11 elvv 5008 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
12 n0 3757 . 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 1370   E.wex 1587    e. wcel 1758    =/= wne 2648   _Vcvv 3078   (/)c0 3748   {csn 3988   <.cop 3994   class class class wbr 4403    X. cxp 4949   dom cdm 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-xp 4957  df-dm 4961
This theorem is referenced by:  rnsnn0  5416  dmsn0  5417  dmsn0el  5419  relsn2  5420  1stnpr  6694  1st2val  6715  mpt2xopxnop0  6845  hashfun  12321
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