Theorem opelcn 11102

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 11094	. . 3 ⊢ ℂ = (R × R)
2	1	eleq2i 2857	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ ⟨𝐴, 𝐵⟩ ∈ (R × R))
3		opelxp 5688	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (R × R) ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R))
4	2, 3	bitri 278	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2145  ⟨cop 4591   × cxp 5650  Rcnr 10838  ℂcc 11086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5168  df-xp 5658  df-c 11094

This theorem is referenced by:  axicn  11123