Theorem opelcn 11167

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 ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R))

Proof of Theorem opelcn

Step	Hyp	Ref	Expression
1		df-c 11159	. . 3 ⊢ ℂ = (R × R)
2	1	eleq2i 2831	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ ⟨𝐴, 𝐵⟩ ∈ (R × R))
3		opelxp 5725	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (R × R) ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R))
4	2, 3	bitri 275	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2106  ⟨cop 4637   × cxp 5687  Rcnr 10903  ℂcc 11151

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-c 11159

This theorem is referenced by:  axicn  11188