Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnm Unicode version

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
Distinct variable group:   ,

Proof of Theorem cnm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxpi 4555 . . 3
2 df-c 7638 . . 3
31, 2eleq2s 2234 . 2
4 vex 2689 . . . . . 6
5 vex 2689 . . . . . 6
6 opm 4156 . . . . . 6
74, 5, 6mpbir2an 926 . . . . 5
8 simprl 520 . . . . . . 7
98eleq2d 2209 . . . . . 6
109exbidv 1797 . . . . 5
117, 10mpbiri 167 . . . 4
1211ex 114 . . 3
1312exlimdvv 1869 . 2
143, 13mpd 13 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wceq 1331  wex 1468   wcel 1480  cvv 2686  cop 3530   cxp 4537  cnr 7117  cc 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
 Copyright terms: Public domain W3C validator