Theorem opelcn 4042

Description: Ordered pair membership in the class of complex numbers.

Hypothesis

Ref	Expression
opelcn.1	⊢ B ∈ V

Assertion

Ref	Expression
opelcn	⊢ (⟨A, B⟩ ∈ ℂ ↔ (A ∈ R ∧ B ∈ R))

Proof of Theorem opelcn

Step	Hyp	Ref	Expression
1		df-c 4034	. . 3 ⊢ ℂ = (R × R)
2	1	eleq2i 1153	. 2 ⊢ (⟨A, B⟩ ∈ ℂ ↔ ⟨A, B⟩ ∈ (R × R))
3		opelcn.1	. . 3 ⊢ B ∈ V
4	3	opelxp 2452	. 2 ⊢ (⟨A, B⟩ ∈ (R × R) ↔ (A ∈ R ∧ B ∈ R))
5	2, 4	bitr 151	1 ⊢ (⟨A, B⟩ ∈ ℂ ↔ (A ∈ R ∧ B ∈ R))

Syntax hints:   ↔ wb 127   ∧ wa 196   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   × cxp 2408  Rcnr 3787  ℂcc 4026

This theorem is referenced by:  axicn 4065  axrecex 4079

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-c 4034

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