Theorem nn0ge2m1nnALT 9997

Description: Alternate proof of nn0ge2m1nn 9606: 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 9906, 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 9606. (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

Step	Hyp	Ref	Expression
1		2z 9651	. . . 4 ⊢ 2 ∈ ℤ
2	1	a1i 9	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 2 ∈ ℤ)
3		nn0z 9643	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
4	3	adantr 276	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 𝑁 ∈ ℤ)
5		simpr 110	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 2 ≤ 𝑁)
6		eluz2 9906	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁))
7	2, 4, 5, 6	syl3anbrc 1212	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 𝑁 ∈ (ℤ≥‘2))
8		uz2m1nn 9984	. 2 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 1) ∈ ℕ)
9	7, 8	syl 14	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2209   class class class wbr 4125  ‘cfv 5372  (class class class)co 6075  1c1 8170   ≤ cle 8351   − cmin 8487  ℕcn 9283  2c2 9334  ℕ₀cn0 9542  ℤcz 9623  ℤ_≥cuz 9900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901

This theorem is referenced by: (None)