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Mirrors > Home > MPE Home > Th. List > opelcn | Structured version Visualization version GIF version |
Description: Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
opelcn | ⊢ (〈𝐴, 𝐵〉 ∈ ℂ ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R)) |
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)) |
Colors of variables: wff setvar class |
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 |
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