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Theorem elz 12508
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 2826 . . 3 (𝑥 = 𝑁 → (𝑥 ∈ ℕ ↔ 𝑁 ∈ ℕ))
3 negeq 11400 . . . 4 (𝑥 = 𝑁 → -𝑥 = -𝑁)
43eleq1d 2823 . . 3 (𝑥 = 𝑁 → (-𝑥 ∈ ℕ ↔ -𝑁 ∈ ℕ))
51, 2, 43orbi123d 1436 . 2 (𝑥 = 𝑁 → ((𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
6 df-z 12507 . 2 ℤ = {𝑥 ∈ ℝ ∣ (𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ)}
75, 6elrab2 3653 1 (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  w3o 1087   = wceq 1542  wcel 2107  cr 11057  0cc0 11058  -cneg 11393  cn 12160  cz 12506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-ov 7365  df-neg 11395  df-z 12507
This theorem is referenced by:  nnnegz  12509  zre  12510  elnnz  12516  0z  12517  elznn0nn  12520  elznn0  12521  elznn  12522  nnz  12527  znegcl  12545  zeo  12596  addmodlteq  13858  zabsle1  26660  ostthlem1  26991  ostth3  27002  elzdif0  32601  qqhval2lem  32602  exp11d  40840
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