Theorem cnm 7798

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 4628	. . 3 |- ( A e. ( R. X. R. ) -> E. u E. v ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) )
2		df-c 7784	. . 3 |- CC = ( R. X. R. )
3	1, 2	eleq2s 2266	. 2 |- ( A e. CC -> E. u E. v ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) )
4		vex 2734	. . . . . 6 |- u e. _V
5		vex 2734	. . . . . 6 |- v e. _V
6		opm 4220	. . . . . 6 |- ( E. x x e. <. u , v >. <-> ( u e. _V /\ v e. _V ) )
7	4, 5, 6	mpbir2an 938	. . . . 5 |- E. x x e. <. u , v >.
8		simprl 527	. . . . . . 7 |- ( ( A e. CC /\ ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) ) -> A = <. u , v >. )
9	8	eleq2d 2241	. . . . . 6 |- ( ( A e. CC /\ ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) ) -> ( x e. A <-> x e. <. u , v >. ) )
10	9	exbidv 1819	. . . . 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 1891	. 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 1349   E.wex 1486   e. wcel 2142   _Vcvv 2731   <.cop 3587   X. cxp 4610   R.cnr 7263   CCcc 7776

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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-v 2733  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-opab 4052  df-xp 4618  df-c 7784

This theorem is referenced by:  axaddf  7834  axmulf  7835