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 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem cnm

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

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)

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