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Mirrors > Home > ILE Home > Th. List > cnm | GIF version |
Description: A complex number is an inhabited set. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by Jim Kingdon, 23-Oct-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnm | ⊢ (𝐴 ∈ ℂ → ∃𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxpi 4555 | . . 3 ⊢ (𝐴 ∈ (R × R) → ∃𝑢∃𝑣(𝐴 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) | |
2 | df-c 7626 | . . 3 ⊢ ℂ = (R × R) | |
3 | 1, 2 | eleq2s 2234 | . 2 ⊢ (𝐴 ∈ ℂ → ∃𝑢∃𝑣(𝐴 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) |
4 | vex 2689 | . . . . . 6 ⊢ 𝑢 ∈ V | |
5 | vex 2689 | . . . . . 6 ⊢ 𝑣 ∈ V | |
6 | opm 4156 | . . . . . 6 ⊢ (∃𝑥 𝑥 ∈ 〈𝑢, 𝑣〉 ↔ (𝑢 ∈ V ∧ 𝑣 ∈ V)) | |
7 | 4, 5, 6 | mpbir2an 926 | . . . . 5 ⊢ ∃𝑥 𝑥 ∈ 〈𝑢, 𝑣〉 |
8 | simprl 520 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) → 𝐴 = 〈𝑢, 𝑣〉) | |
9 | 8 | eleq2d 2209 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 〈𝑢, 𝑣〉)) |
10 | 9 | exbidv 1797 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) → (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 〈𝑢, 𝑣〉)) |
11 | 7, 10 | mpbiri 167 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) → ∃𝑥 𝑥 ∈ 𝐴) |
12 | 11 | ex 114 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R)) → ∃𝑥 𝑥 ∈ 𝐴)) |
13 | 12 | exlimdvv 1869 | . 2 ⊢ (𝐴 ∈ ℂ → (∃𝑢∃𝑣(𝐴 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R)) → ∃𝑥 𝑥 ∈ 𝐴)) |
14 | 3, 13 | mpd 13 | 1 ⊢ (𝐴 ∈ ℂ → ∃𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Vcvv 2686 〈cop 3530 × cxp 4537 Rcnr 7105 ℂcc 7618 |
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 2121 ax-sep 4046 ax-pow 4098 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-xp 4545 df-c 7626 |
This theorem is referenced by: axaddf 7676 axmulf 7677 |
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