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Theorem cnm 7652
Description: A complex number is an inhabited set. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by Jim Kingdon, 23-Oct-2023.) (New usage is discouraged.)
Assertion
Ref Expression
cnm  |-  ( A  e.  CC  ->  E. x  x  e.  A )
Distinct variable group:    x, A

Proof of Theorem cnm
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxpi 4555 . . 3  |-  ( A  e.  ( R.  X.  R. )  ->  E. u E. v ( A  = 
<. u ,  v >.  /\  ( u  e.  R.  /\  v  e.  R. )
) )
2 df-c 7638 . . 3  |-  CC  =  ( R.  X.  R. )
31, 2eleq2s 2234 . 2  |-  ( A  e.  CC  ->  E. u E. v ( A  = 
<. u ,  v >.  /\  ( u  e.  R.  /\  v  e.  R. )
) )
4 vex 2689 . . . . . 6  |-  u  e. 
_V
5 vex 2689 . . . . . 6  |-  v  e. 
_V
6 opm 4156 . . . . . 6  |-  ( E. x  x  e.  <. u ,  v >.  <->  ( u  e.  _V  /\  v  e. 
_V ) )
74, 5, 6mpbir2an 926 . . . . 5  |-  E. x  x  e.  <. u ,  v >.
8 simprl 520 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( A  =  <. u ,  v >.  /\  (
u  e.  R.  /\  v  e.  R. )
) )  ->  A  =  <. u ,  v
>. )
98eleq2d 2209 . . . . . 6  |-  ( ( A  e.  CC  /\  ( A  =  <. u ,  v >.  /\  (
u  e.  R.  /\  v  e.  R. )
) )  ->  (
x  e.  A  <->  x  e.  <.
u ,  v >.
) )
109exbidv 1797 . . . . 5  |-  ( ( A  e.  CC  /\  ( A  =  <. u ,  v >.  /\  (
u  e.  R.  /\  v  e.  R. )
) )  ->  ( E. x  x  e.  A 
<->  E. x  x  e. 
<. u ,  v >.
) )
117, 10mpbiri 167 . . . 4  |-  ( ( A  e.  CC  /\  ( A  =  <. u ,  v >.  /\  (
u  e.  R.  /\  v  e.  R. )
) )  ->  E. x  x  e.  A )
1211ex 114 . . 3  |-  ( A  e.  CC  ->  (
( A  =  <. u ,  v >.  /\  (
u  e.  R.  /\  v  e.  R. )
)  ->  E. x  x  e.  A )
)
1312exlimdvv 1869 . 2  |-  ( A  e.  CC  ->  ( E. u E. v ( A  =  <. u ,  v >.  /\  (
u  e.  R.  /\  v  e.  R. )
)  ->  E. x  x  e.  A )
)
143, 13mpd 13 1  |-  ( A  e.  CC  ->  E. x  x  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331   E.wex 1468    e. wcel 1480   _Vcvv 2686   <.cop 3530    X. cxp 4537   R.cnr 7117   CCcc 7630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545  df-c 7638
This theorem is referenced by:  axaddf  7688  axmulf  7689
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