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Theorem opelcn 7839
Description: Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.)
Assertion
Ref Expression
opelcn (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ (𝐴R𝐵R))

Proof of Theorem opelcn
StepHypRef Expression
1 df-c 7831 . . 3 ℂ = (R × R)
21eleq2i 2254 . 2 (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ ⟨𝐴, 𝐵⟩ ∈ (R × R))
3 opelxp 4668 . 2 (⟨𝐴, 𝐵⟩ ∈ (R × R) ↔ (𝐴R𝐵R))
42, 3bitri 184 1 (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ (𝐴R𝐵R))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wcel 2158  cop 3607   × cxp 4636  Rcnr 7310  cc 7823
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-opab 4077  df-xp 4644  df-c 7831
This theorem is referenced by:  axicn  7876
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