Theorem cnm 7754

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 4604	. . 3 ⊢ (𝐴 ∈ (R × R) → ∃𝑢∃𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R)))
2		df-c 7740	. . 3 ⊢ ℂ = (R × R)
3	1, 2	eleq2s 2252	. 2 ⊢ (𝐴 ∈ ℂ → ∃𝑢∃𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R)))
4		vex 2715	. . . . . 6 ⊢ 𝑢 ∈ V
5		vex 2715	. . . . . 6 ⊢ 𝑣 ∈ V
6		opm 4196	. . . . . 6 ⊢ (∃𝑥 𝑥 ∈ ⟨𝑢, 𝑣⟩ ↔ (𝑢 ∈ V ∧ 𝑣 ∈ V))
7	4, 5, 6	mpbir2an 927	. . . . 5 ⊢ ∃𝑥 𝑥 ∈ ⟨𝑢, 𝑣⟩
8		simprl 521	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) → 𝐴 = ⟨𝑢, 𝑣⟩)
9	8	eleq2d 2227	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⟨𝑢, 𝑣⟩))
10	9	exbidv 1805	. . . . 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 1877	. 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 1335  ∃wex 1472   ∈ wcel 2128  Vcvv 2712  ⟨cop 3564   × cxp 4586  Rcnr 7219  ℂcc 7732

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-opab 4028  df-xp 4594  df-c 7740

This theorem is referenced by:  axaddf  7790  axmulf  7791