Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7619 | . . 3 ⊢ ℂ = (R × R) | |
2 | 1 | eleq2i 2204 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ ℂ ↔ 〈𝐴, 𝐵〉 ∈ (R × R)) |
3 | opelxp 4564 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (R × R) ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R)) | |
4 | 2, 3 | bitri 183 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ ℂ ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1480 〈cop 3525 × cxp 4532 Rcnr 7098 ℂcc 7611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 df-xp 4540 df-c 7619 |
This theorem is referenced by: axicn 7664 |
Copyright terms: Public domain | W3C validator |