Theorem cnm 7773

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 4620	. . 3 |- ( A e. ( R. X. R. ) -> E. u E. v ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) )
2		df-c 7759	. . 3 |- CC = ( R. X. R. )
3	1, 2	eleq2s 2261	. 2 |- ( A e. CC -> E. u E. v ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) )
4		vex 2729	. . . . . 6 |- u e. _V
5		vex 2729	. . . . . 6 |- v e. _V
6		opm 4212	. . . . . 6 |- ( E. x x e. <. u , v >. <-> ( u e. _V /\ v e. _V ) )
7	4, 5, 6	mpbir2an 932	. . . . 5 |- E. x x e. <. u , v >.
8		simprl 521	. . . . . . 7 |- ( ( A e. CC /\ ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) ) -> A = <. u , v >. )
9	8	eleq2d 2236	. . . . . 6 |- ( ( A e. CC /\ ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) ) -> ( x e. A <-> x e. <. u , v >. ) )
10	9	exbidv 1813	. . . . 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 1885	. 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 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   <.cop 3579   X. cxp 4602   R.cnr 7238   CCcc 7751

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044  df-xp 4610  df-c 7759

This theorem is referenced by:  axaddf  7809  axmulf  7810