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Theorem elz 8247
 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 2046 . . 3 (𝑥 = 𝑁 → (𝑥 = 0 ↔ 𝑁 = 0))
2 eleq1 2100 . . 3 (𝑥 = 𝑁 → (𝑥 ∈ ℕ ↔ 𝑁 ∈ ℕ))
3 negeq 7204 . . . 4 (𝑥 = 𝑁 → -𝑥 = -𝑁)
43eleq1d 2106 . . 3 (𝑥 = 𝑁 → (-𝑥 ∈ ℕ ↔ -𝑁 ∈ ℕ))
51, 2, 43orbi123d 1206 . 2 (𝑥 = 𝑁 → ((𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
6 df-z 8246 . 2 ℤ = {𝑥 ∈ ℝ ∣ (𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ)}
75, 6elrab2 2700 1 (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∨ w3o 884   = wceq 1243   ∈ wcel 1393  ℝcr 6888  0cc0 6889  -cneg 7183  ℕcn 7914  ℤcz 8245 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515  df-neg 7185  df-z 8246 This theorem is referenced by:  nnnegz  8248  zre  8249  elnnz  8255  0z  8256  elnn0z  8258  elznn0nn  8259  elznn0  8260  elznn  8261  znegcl  8276  zaddcl  8285  ztri3or0  8287  zeo  8343
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