MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opelcn Structured version   Visualization version   GIF version

Theorem opelcn 10550
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⟩ ∈ ℂ ↔ (𝐴R𝐵R))

Proof of Theorem opelcn
StepHypRef Expression
1 df-c 10542 . . 3 ℂ = (R × R)
21eleq2i 2904 . 2 (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ ⟨𝐴, 𝐵⟩ ∈ (R × R))
3 opelxp 5590 . 2 (⟨𝐴, 𝐵⟩ ∈ (R × R) ↔ (𝐴R𝐵R))
42, 3bitri 277 1 (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ (𝐴R𝐵R))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  wcel 2110  cop 4572   × cxp 5552  Rcnr 10286  cc 10534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-opab 5128  df-xp 5560  df-c 10542
This theorem is referenced by:  axicn  10571
  Copyright terms: Public domain W3C validator