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Theorem elz 12613
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 2739 . . 3 (𝑥 = 𝑁 → (𝑥 = 0 ↔ 𝑁 = 0))
2 eleq1 2827 . . 3 (𝑥 = 𝑁 → (𝑥 ∈ ℕ ↔ 𝑁 ∈ ℕ))
3 negeq 11498 . . . 4 (𝑥 = 𝑁 → -𝑥 = -𝑁)
43eleq1d 2824 . . 3 (𝑥 = 𝑁 → (-𝑥 ∈ ℕ ↔ -𝑁 ∈ ℕ))
51, 2, 43orbi123d 1434 . 2 (𝑥 = 𝑁 → ((𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
6 df-z 12612 . 2 ℤ = {𝑥 ∈ ℝ ∣ (𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ)}
75, 6elrab2 3698 1 (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3o 1085   = wceq 1537  wcel 2106  cr 11152  0cc0 11153  -cneg 11491  cn 12264  cz 12611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-neg 11493  df-z 12612
This theorem is referenced by:  nnnegz  12614  zre  12615  elnnz  12621  0z  12622  elznn0nn  12625  elznn0  12626  elznn  12627  nnz  12632  znegcl  12650  zeo  12702  addmodlteq  13984  zabsle1  27355  ostthlem1  27686  ostth3  27697  elzdif0  33943  qqhval2lem  33944  exp11d  42340
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