Theorem cnm 7975

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 4704	. . 3 ⊢ (𝐴 ∈ (R × R) → ∃𝑢∃𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R)))
2		df-c 7961	. . 3 ⊢ ℂ = (R × R)
3	1, 2	eleq2s 2301	. 2 ⊢ (𝐴 ∈ ℂ → ∃𝑢∃𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R)))
4		vex 2776	. . . . . 6 ⊢ 𝑢 ∈ V
5		vex 2776	. . . . . 6 ⊢ 𝑣 ∈ V
6		opm 4291	. . . . . 6 ⊢ (∃𝑥 𝑥 ∈ ⟨𝑢, 𝑣⟩ ↔ (𝑢 ∈ V ∧ 𝑣 ∈ V))
7	4, 5, 6	mpbir2an 945	. . . . 5 ⊢ ∃𝑥 𝑥 ∈ ⟨𝑢, 𝑣⟩
8		simprl 529	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) → 𝐴 = ⟨𝑢, 𝑣⟩)
9	8	eleq2d 2276	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ R ∧ 𝑣 ∈ R))) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⟨𝑢, 𝑣⟩))
10	9	exbidv 1849	. . . . 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 1922	. 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 1373  ∃wex 1516   ∈ wcel 2177  Vcvv 2773  ⟨cop 3641   × cxp 4686  Rcnr 7440  ℂcc 7953

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-opab 4117  df-xp 4694  df-c 7961

This theorem is referenced by:  axaddf  8011  axmulf  8012