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Mirrors > Home > ILE Home > Th. List > opelcn | GIF version |
Description: Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) |
Ref | Expression |
---|---|
opelcn | ⊢ (〈𝐴, 𝐵〉 ∈ ℂ ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-c 7831 | . . 3 ⊢ ℂ = (R × R) | |
2 | 1 | eleq2i 2254 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ℂ ↔ 〈𝐴, 𝐵〉 ∈ (R × R)) |
3 | opelxp 4668 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (R × R) ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R)) | |
4 | 2, 3 | bitri 184 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ ℂ ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∈ wcel 2158 〈cop 3607 × cxp 4636 Rcnr 7310 ℂcc 7823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-opab 4077 df-xp 4644 df-c 7831 |
This theorem is referenced by: axicn 7876 |
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