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Theorem elz 11324
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 2630 . . 3 (𝑥 = 𝑁 → (𝑥 = 0 ↔ 𝑁 = 0))
2 eleq1 2692 . . 3 (𝑥 = 𝑁 → (𝑥 ∈ ℕ ↔ 𝑁 ∈ ℕ))
3 negeq 10218 . . . 4 (𝑥 = 𝑁 → -𝑥 = -𝑁)
43eleq1d 2688 . . 3 (𝑥 = 𝑁 → (-𝑥 ∈ ℕ ↔ -𝑁 ∈ ℕ))
51, 2, 43orbi123d 1395 . 2 (𝑥 = 𝑁 → ((𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
6 df-z 11323 . 2 ℤ = {𝑥 ∈ ℝ ∣ (𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ)}
75, 6elrab2 3353 1 (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3o 1035   = wceq 1480  wcel 1992  cr 9880  0cc0 9881  -cneg 10212  cn 10965  cz 11322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5813  df-fv 5858  df-ov 6608  df-neg 10214  df-z 11323
This theorem is referenced by:  nnnegz  11325  zre  11326  elnnz  11332  0z  11333  elznn0nn  11336  elznn0  11337  elznn  11338  znegcl  11357  zeo  11407  addmodlteq  12682  zabsle1  24916  ostthlem1  25211  ostth3  25222  elzdif0  29798  qqhval2lem  29799
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