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Theorem elz 12615
Description: Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
Assertion
Ref Expression
elz (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))

Proof of Theorem elz
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2741 . . 3 (𝑥 = 𝑁 → (𝑥 = 0 ↔ 𝑁 = 0))
2 eleq1 2829 . . 3 (𝑥 = 𝑁 → (𝑥 ∈ ℕ ↔ 𝑁 ∈ ℕ))
3 negeq 11500 . . . 4 (𝑥 = 𝑁 → -𝑥 = -𝑁)
43eleq1d 2826 . . 3 (𝑥 = 𝑁 → (-𝑥 ∈ ℕ ↔ -𝑁 ∈ ℕ))
51, 2, 43orbi123d 1437 . 2 (𝑥 = 𝑁 → ((𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
6 df-z 12614 . 2 ℤ = {𝑥 ∈ ℝ ∣ (𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ)}
75, 6elrab2 3695 1 (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3o 1086   = wceq 1540  wcel 2108  cr 11154  0cc0 11155  -cneg 11493  cn 12266  cz 12613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-neg 11495  df-z 12614
This theorem is referenced by:  nnnegz  12616  zre  12617  elnnz  12623  0z  12624  elznn0nn  12627  elznn0  12628  elznn  12629  nnz  12634  znegcl  12652  zeo  12704  addmodlteq  13987  zabsle1  27340  ostthlem1  27671  ostth3  27682  elzdif0  33981  qqhval2lem  33982  exp11d  42361
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