Theorem cnm 7640

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.

Step	Hyp	Ref	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 7626	. . 3 |- CC = ( R. X. R. )
3	1, 2	eleq2s 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 ) )
7	4, 5, 6	mpbir2an 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 >. )
9	8	eleq2d 2209	. . . . . 6 |- ( ( A e. CC /\ ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) ) -> ( x e. A <-> x e. <. u , v >. ) )
10	9	exbidv 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 >. ) )
11	7, 10	mpbiri 167	. . . 4 |- ( ( A e. CC /\ ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) ) -> E. x x e. A )
12	11	ex 114	. . 3 |- ( A e. CC -> ( ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) -> E. x x e. A ) )
13	12	exlimdvv 1869	. 2 |- ( A e. CC -> ( E. u E. v ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) -> E. x x e. A ) )
14	3, 13	mpd 13	1 |- ( A e. CC -> E. x x e. A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   _Vcvv 2686   <.cop 3530   X. cxp 4537   R.cnr 7105   CCcc 7618

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 7626

This theorem is referenced by:  axaddf  7676  axmulf  7677