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Theorem nn1m1nn 8007
Description: Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
Assertion
Ref Expression
nn1m1nn (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))

Proof of Theorem nn1m1nn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 orc 643 . . 3 (𝑥 = 1 → (𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ))
2 1cnd 7100 . . 3 (𝑥 = 1 → 1 ∈ ℂ)
31, 22thd 168 . 2 (𝑥 = 1 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ 1 ∈ ℂ))
4 eqeq1 2062 . . 3 (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1))
5 oveq1 5546 . . . 4 (𝑥 = 𝑦 → (𝑥 − 1) = (𝑦 − 1))
65eleq1d 2122 . . 3 (𝑥 = 𝑦 → ((𝑥 − 1) ∈ ℕ ↔ (𝑦 − 1) ∈ ℕ))
74, 6orbi12d 717 . 2 (𝑥 = 𝑦 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ (𝑦 = 1 ∨ (𝑦 − 1) ∈ ℕ)))
8 eqeq1 2062 . . 3 (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1))
9 oveq1 5546 . . . 4 (𝑥 = (𝑦 + 1) → (𝑥 − 1) = ((𝑦 + 1) − 1))
109eleq1d 2122 . . 3 (𝑥 = (𝑦 + 1) → ((𝑥 − 1) ∈ ℕ ↔ ((𝑦 + 1) − 1) ∈ ℕ))
118, 10orbi12d 717 . 2 (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ ((𝑦 + 1) = 1 ∨ ((𝑦 + 1) − 1) ∈ ℕ)))
12 eqeq1 2062 . . 3 (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1))
13 oveq1 5546 . . . 4 (𝑥 = 𝐴 → (𝑥 − 1) = (𝐴 − 1))
1413eleq1d 2122 . . 3 (𝑥 = 𝐴 → ((𝑥 − 1) ∈ ℕ ↔ (𝐴 − 1) ∈ ℕ))
1512, 14orbi12d 717 . 2 (𝑥 = 𝐴 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ)))
16 ax-1cn 7034 . 2 1 ∈ ℂ
17 nncn 7997 . . . . . 6 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
18 pncan 7279 . . . . . 6 ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑦 + 1) − 1) = 𝑦)
1917, 16, 18sylancl 398 . . . . 5 (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) = 𝑦)
20 id 19 . . . . 5 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ)
2119, 20eqeltrd 2130 . . . 4 (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) ∈ ℕ)
2221olcd 663 . . 3 (𝑦 ∈ ℕ → ((𝑦 + 1) = 1 ∨ ((𝑦 + 1) − 1) ∈ ℕ))
2322a1d 22 . 2 (𝑦 ∈ ℕ → ((𝑦 = 1 ∨ (𝑦 − 1) ∈ ℕ) → ((𝑦 + 1) = 1 ∨ ((𝑦 + 1) − 1) ∈ ℕ)))
243, 7, 11, 15, 16, 23nnind 8005 1 (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 639   = wceq 1259  wcel 1409  (class class class)co 5539  cc 6944  1c1 6947   + caddc 6949  cmin 7244  cn 7989
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-in1 554  ax-in2 555  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-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-setind 4289  ax-cnex 7032  ax-resscn 7033  ax-1cn 7034  ax-1re 7035  ax-icn 7036  ax-addcl 7037  ax-addrcl 7038  ax-mulcl 7039  ax-addcom 7041  ax-addass 7043  ax-distr 7045  ax-i2m1 7046  ax-0id 7049  ax-rnegex 7050  ax-cnre 7052
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-iota 4894  df-fun 4931  df-fv 4937  df-riota 5495  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-sub 7246  df-inn 7990
This theorem is referenced by:  nn1suc  8008  nnsub  8027  nnm1nn0  8279  nn0ge2m1nn  8298
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