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Theorem opelcn 8767
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 8759 . . 3  |-  CC  =  ( R.  X.  R. )
21eleq2i 2360 . 2  |-  ( <. A ,  B >.  e.  CC  <->  <. A ,  B >.  e.  ( R.  X.  R. ) )
3 opelxp 4735 . 2  |-  ( <. A ,  B >.  e.  ( R.  X.  R. ) 
<->  ( A  e.  R.  /\  B  e.  R. )
)
42, 3bitri 240 1  |-  ( <. A ,  B >.  e.  CC  <->  ( A  e. 
R.  /\  B  e.  R. ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1696   <.cop 3656    X. cxp 4703   R.cnr 8505   CCcc 8751
This theorem is referenced by:  axicn  8788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711  df-c 8759
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