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Theorem opelcn 8996
Description: Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
opelcn  |-  ( <. A ,  B >.  e.  CC  <->  ( A  e. 
R.  /\  B  e.  R. ) )

Proof of Theorem opelcn
StepHypRef Expression
1 df-c 8988 . . 3  |-  CC  =  ( R.  X.  R. )
21eleq2i 2499 . 2  |-  ( <. A ,  B >.  e.  CC  <->  <. A ,  B >.  e.  ( R.  X.  R. ) )
3 opelxp 4900 . 2  |-  ( <. A ,  B >.  e.  ( R.  X.  R. ) 
<->  ( A  e.  R.  /\  B  e.  R. )
)
42, 3bitri 241 1  |-  ( <. A ,  B >.  e.  CC  <->  ( A  e. 
R.  /\  B  e.  R. ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1725   <.cop 3809    X. cxp 4868   R.cnr 8734   CCcc 8980
This theorem is referenced by:  axicn  9017
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259  df-xp 4876  df-c 8988
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