Theorem opelcn 10886

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 10878	. . 3 ⊢ ℂ = (R × R)
2	1	eleq2i 2832	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ ⟨𝐴, 𝐵⟩ ∈ (R × R))
3		opelxp 5626	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (R × R) ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R))
4	2, 3	bitri 274	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∈ wcel 2110  ⟨cop 4573   × cxp 5588  Rcnr 10622  ℂcc 10870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5142  df-xp 5596  df-c 10878

This theorem is referenced by:  axicn  10907