Theorem cnm 8051

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 4741	. . 3 |- ( A e. ( R. X. R. ) -> E. u E. v ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) )
2		df-c 8037	. . 3 |- CC = ( R. X. R. )
3	1, 2	eleq2s 2326	. 2 |- ( A e. CC -> E. u E. v ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) )
4		vex 2805	. . . . . 6 |- u e. _V
5		vex 2805	. . . . . 6 |- v e. _V
6		opm 4326	. . . . . 6 |- ( E. x x e. <. u , v >. <-> ( u e. _V /\ v e. _V ) )
7	4, 5, 6	mpbir2an 950	. . . . 5 |- E. x x e. <. u , v >.
8		simprl 531	. . . . . . 7 |- ( ( A e. CC /\ ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) ) -> A = <. u , v >. )
9	8	eleq2d 2301	. . . . . 6 |- ( ( A e. CC /\ ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) ) -> ( x e. A <-> x e. <. u , v >. ) )
10	9	exbidv 1873	. . . . 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 168	. . . 4 |- ( ( A e. CC /\ ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) ) -> E. x x e. A )
12	11	ex 115	. . 3 |- ( A e. CC -> ( ( A = <. u , v >. /\ ( u e. R. /\ v e. R. ) ) -> E. x x e. A ) )
13	12	exlimdvv 1946	. 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 104   = wceq 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802   <.cop 3672   X. cxp 4723   R.cnr 7516   CCcc 8029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-xp 4731  df-c 8037

This theorem is referenced by:  axaddf  8087  axmulf  8088