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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 4627 | . . 3 ⊢ (𝐴 ∈ (R × R) → ∃𝑢∃𝑣(𝐴 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) | |
2 | df-c 7780 | . . 3 ⊢ ℂ = (R × R) | |
3 | 1, 2 | eleq2s 2265 | . 2 ⊢ (𝐴 ∈ ℂ → ∃𝑢∃𝑣(𝐴 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) |
4 | vex 2733 | . . . . . 6 ⊢ 𝑢 ∈ V | |
5 | vex 2733 | . . . . . 6 ⊢ 𝑣 ∈ V | |
6 | opm 4219 | . . . . . 6 ⊢ (∃𝑥 𝑥 ∈ 〈𝑢, 𝑣〉 ↔ (𝑢 ∈ V ∧ 𝑣 ∈ V)) | |
7 | 4, 5, 6 | mpbir2an 937 | . . . . 5 ⊢ ∃𝑥 𝑥 ∈ 〈𝑢, 𝑣〉 |
8 | simprl 526 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) → 𝐴 = 〈𝑢, 𝑣〉) | |
9 | 8 | eleq2d 2240 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 = 〈𝑢, 𝑣〉 ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 〈𝑢, 𝑣〉)) |
10 | 9 | exbidv 1818 | . . . . 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 1890 | . 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 1348 ∃wex 1485 ∈ wcel 2141 Vcvv 2730 〈cop 3586 × cxp 4609 Rcnr 7259 ℂcc 7772 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-opab 4051 df-xp 4617 df-c 7780 |
This theorem is referenced by: axaddf 7830 axmulf 7831 |
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