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Theorem opelcn 6960
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 6952 . . 3 ℂ = (R × R)
21eleq2i 2120 . 2 (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ ⟨𝐴, 𝐵⟩ ∈ (R × R))
3 opelxp 4401 . 2 (⟨𝐴, 𝐵⟩ ∈ (R × R) ↔ (𝐴R𝐵R))
42, 3bitri 177 1 (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ (𝐴R𝐵R))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wcel 1409  cop 3405   × cxp 4370  Rcnr 6452  cc 6944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-opab 3846  df-xp 4378  df-c 6952
This theorem is referenced by:  axicn  6996
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