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Theorem opelcn 5220
Description: Ordered pair membership in the class of complex numbers.
Hypothesis
Ref Expression
opelcn.1 |- B e. V
Assertion
Ref Expression
opelcn |- (<.A, B>. e. CC <-> (A e. R. /\ B e. R.))
Proof of Theorem opelcn
StepHypRef Expression
1 df-c 5212 . . 3 |- CC = (R. X. R.)
21eleq2i 1530 . 2 |- (<.A, B>. e. CC <-> <.A, B>. e. (R. X. R.))
3 opelcn.1 . . 3 |- B e. V
43opelxp 3204 . 2 |- (<.A, B>. e. (R. X. R.) <-> (A e. R. /\ B e. R.))
52, 4bitr 173 1 |- (<.A, B>. e. CC <-> (A e. R. /\ B e. R.))
Colors of variables: wff set class
Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 955  Vcvv 1802  <.cop 2401   X. cxp 3158  R.cnr 4965  CCcc 5204
This theorem is referenced by:  axicn 5242
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-opab 2657  df-xp 3174  df-c 5212
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