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Theorem elnn0 8240
 Description: Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
Assertion
Ref Expression
elnn0 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))

Proof of Theorem elnn0
StepHypRef Expression
1 df-n0 8239 . . 3 0 = (ℕ ∪ {0})
21eleq2i 2120 . 2 (𝐴 ∈ ℕ0𝐴 ∈ (ℕ ∪ {0}))
3 elun 3111 . 2 (𝐴 ∈ (ℕ ∪ {0}) ↔ (𝐴 ∈ ℕ ∨ 𝐴 ∈ {0}))
4 c0ex 7078 . . . 4 0 ∈ V
54elsn2 3432 . . 3 (𝐴 ∈ {0} ↔ 𝐴 = 0)
65orbi2i 689 . 2 ((𝐴 ∈ ℕ ∨ 𝐴 ∈ {0}) ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
72, 3, 63bitri 199 1 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
 Colors of variables: wff set class Syntax hints:   ↔ wb 102   ∨ wo 639   = wceq 1259   ∈ wcel 1409   ∪ cun 2942  {csn 3402  0cc0 6946  ℕcn 7989  ℕ0cn0 8238 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-1cn 7034  ax-icn 7036  ax-addcl 7037  ax-mulcl 7039  ax-i2m1 7046 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-sn 3408  df-n0 8239 This theorem is referenced by:  0nn0  8253  nn0ge0  8263  nnnn0addcl  8268  nnm1nn0  8279  elnnnn0b  8282  elnn0z  8314  elznn0nn  8315  elznn0  8316  elznn  8317  nn0ind-raph  8413  nn0ledivnn  8784  expp1  9421  expnegap0  9422  expcllem  9425  facp1  9591  faclbnd  9602  faclbnd3  9604  bcn1  9619  ibcval5  9624  nn0enne  10206  nn0o1gt2  10209
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