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Theorem cnm 7794
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.)
Assertion
Ref Expression
cnm (𝐴 ∈ ℂ → ∃𝑥 𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem cnm
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxpi 4627 . . 3 (𝐴 ∈ (R × R) → ∃𝑢𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢R𝑣R)))
2 df-c 7780 . . 3 ℂ = (R × R)
31, 2eleq2s 2265 . 2 (𝐴 ∈ ℂ → ∃𝑢𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢R𝑣R)))
4 vex 2733 . . . . . 6 𝑢 ∈ V
5 vex 2733 . . . . . 6 𝑣 ∈ V
6 opm 4219 . . . . . 6 (∃𝑥 𝑥 ∈ ⟨𝑢, 𝑣⟩ ↔ (𝑢 ∈ V ∧ 𝑣 ∈ V))
74, 5, 6mpbir2an 937 . . . . 5 𝑥 𝑥 ∈ ⟨𝑢, 𝑣
8 simprl 526 . . . . . . 7 ((𝐴 ∈ ℂ ∧ (𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢R𝑣R))) → 𝐴 = ⟨𝑢, 𝑣⟩)
98eleq2d 2240 . . . . . 6 ((𝐴 ∈ ℂ ∧ (𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢R𝑣R))) → (𝑥𝐴𝑥 ∈ ⟨𝑢, 𝑣⟩))
109exbidv 1818 . . . . 5 ((𝐴 ∈ ℂ ∧ (𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢R𝑣R))) → (∃𝑥 𝑥𝐴 ↔ ∃𝑥 𝑥 ∈ ⟨𝑢, 𝑣⟩))
117, 10mpbiri 167 . . . 4 ((𝐴 ∈ ℂ ∧ (𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢R𝑣R))) → ∃𝑥 𝑥𝐴)
1211ex 114 . . 3 (𝐴 ∈ ℂ → ((𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢R𝑣R)) → ∃𝑥 𝑥𝐴))
1312exlimdvv 1890 . 2 (𝐴 ∈ ℂ → (∃𝑢𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢R𝑣R)) → ∃𝑥 𝑥𝐴))
143, 13mpd 13 1 (𝐴 ∈ ℂ → ∃𝑥 𝑥𝐴)
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  cc 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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