Theorem opelcn 11198

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 11190	. . 3 ⊢ ℂ = (R × R)
2	1	eleq2i 2836	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ℂ ↔ ⟨𝐴, 𝐵⟩ ∈ (R × R))
3		opelxp 5736	. 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 2108  ⟨cop 4654   × cxp 5698  Rcnr 10934  ℂcc 11182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-xp 5706  df-c 11190

This theorem is referenced by:  axicn  11219