Theorem cnm 8189

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 4785	. . 3 ⊢ (𝐴 ∈ (R × R) → ∃𝑢∃𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R)))
2		df-c 8175	. . 3 ⊢ ℂ = (R × R)
3	1, 2	eleq2s 2333	. 2 ⊢ (𝐴 ∈ ℂ → ∃𝑢∃𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R)))
4		vex 2824	. . . . . 6 ⊢ 𝑢 ∈ V
5		vex 2824	. . . . . 6 ⊢ 𝑣 ∈ V
6		opm 4369	. . . . . 6 ⊢ (∃𝑥 𝑥 ∈ ⟨𝑢, 𝑣⟩ ↔ (𝑢 ∈ V ∧ 𝑣 ∈ V))
7	4, 5, 6	mpbir2an 955	. . . . 5 ⊢ ∃𝑥 𝑥 ∈ ⟨𝑢, 𝑣⟩
8		simprl 535	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) → 𝐴 = ⟨𝑢, 𝑣⟩)
9	8	eleq2d 2308	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⟨𝑢, 𝑣⟩))
10	9	exbidv 1878	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) → (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ ⟨𝑢, 𝑣⟩))
11	7, 10	mpbiri 168	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) → ∃𝑥 𝑥 ∈ 𝐴)
12	11	ex 115	. . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R)) → ∃𝑥 𝑥 ∈ 𝐴))
13	12	exlimdvv 1953	. 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 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821  ⟨cop 3708   × cxp 4767  Rcnr 7654  ℂcc 8167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-opab 4188  df-xp 4775  df-c 8175

This theorem is referenced by:  axaddf  8225  axmulf  8226