Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  nn0ge2m1nnALT GIF version

Theorem nn0ge2m1nnALT 9164
 Description: Alternate proof of nn0ge2m1nn 8794: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. This version is proved using eluz2 9086, a theorem for upper sets of integers, which are defined later than the positive and nonnegative integers. This proof is, however, much shorter than the proof of nn0ge2m1nn 8794. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
nn0ge2m1nnALT ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)

Proof of Theorem nn0ge2m1nnALT
StepHypRef Expression
1 2z 8839 . . . 4 2 ∈ ℤ
21a1i 9 . . 3 ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 2 ∈ ℤ)
3 nn0z 8831 . . . 4 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
43adantr 271 . . 3 ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 𝑁 ∈ ℤ)
5 simpr 109 . . 3 ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 2 ≤ 𝑁)
6 eluz2 9086 . . 3 (𝑁 ∈ (ℤ‘2) ↔ (2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁))
72, 4, 5, 6syl3anbrc 1128 . 2 ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 𝑁 ∈ (ℤ‘2))
8 uz2m1nn 9153 . 2 (𝑁 ∈ (ℤ‘2) → (𝑁 − 1) ∈ ℕ)
97, 8syl 14 1 ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1439   class class class wbr 3851  ‘cfv 5028  (class class class)co 5666  1c1 7412   ≤ cle 7584   − cmin 7714  ℕcn 8483  2c2 8534  ℕ0cn0 8734  ℤcz 8811  ℤ≥cuz 9080 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-ltadd 7522 This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-inn 8484  df-2 8542  df-n0 8735  df-z 8812  df-uz 9081 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator