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Theorem nn1gt1 8846
 Description: A positive integer is either one or greater than one. This is for ℕ; 0elnn 4572 is a similar theorem for ω (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.)
Assertion
Ref Expression
nn1gt1 (𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴))

Proof of Theorem nn1gt1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2161 . . 3 (𝑥 = 1 → (𝑥 = 1 ↔ 1 = 1))
2 breq2 3965 . . 3 (𝑥 = 1 → (1 < 𝑥 ↔ 1 < 1))
31, 2orbi12d 783 . 2 (𝑥 = 1 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (1 = 1 ∨ 1 < 1)))
4 eqeq1 2161 . . 3 (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1))
5 breq2 3965 . . 3 (𝑥 = 𝑦 → (1 < 𝑥 ↔ 1 < 𝑦))
64, 5orbi12d 783 . 2 (𝑥 = 𝑦 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝑦 = 1 ∨ 1 < 𝑦)))
7 eqeq1 2161 . . 3 (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1))
8 breq2 3965 . . 3 (𝑥 = (𝑦 + 1) → (1 < 𝑥 ↔ 1 < (𝑦 + 1)))
97, 8orbi12d 783 . 2 (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1))))
10 eqeq1 2161 . . 3 (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1))
11 breq2 3965 . . 3 (𝑥 = 𝐴 → (1 < 𝑥 ↔ 1 < 𝐴))
1210, 11orbi12d 783 . 2 (𝑥 = 𝐴 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝐴 = 1 ∨ 1 < 𝐴)))
13 eqid 2154 . . 3 1 = 1
1413orci 721 . 2 (1 = 1 ∨ 1 < 1)
15 nngt0 8837 . . . . 5 (𝑦 ∈ ℕ → 0 < 𝑦)
16 nnre 8819 . . . . . 6 (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
17 1re 7856 . . . . . 6 1 ∈ ℝ
18 ltaddpos2 8307 . . . . . 6 ((𝑦 ∈ ℝ ∧ 1 ∈ ℝ) → (0 < 𝑦 ↔ 1 < (𝑦 + 1)))
1916, 17, 18sylancl 410 . . . . 5 (𝑦 ∈ ℕ → (0 < 𝑦 ↔ 1 < (𝑦 + 1)))
2015, 19mpbid 146 . . . 4 (𝑦 ∈ ℕ → 1 < (𝑦 + 1))
2120olcd 724 . . 3 (𝑦 ∈ ℕ → ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1)))
2221a1d 22 . 2 (𝑦 ∈ ℕ → ((𝑦 = 1 ∨ 1 < 𝑦) → ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1))))
233, 6, 9, 12, 14, 22nnind 8828 1 (𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 698   = wceq 1332   ∈ wcel 2125   class class class wbr 3961  (class class class)co 5814  ℝcr 7710  0cc0 7711  1c1 7712   + caddc 7714   < clt 7891  ℕcn 8812 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-addcom 7811  ax-addass 7813  ax-i2m1 7816  ax-0lt1 7817  ax-0id 7819  ax-rnegex 7820  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-ltadd 7827 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-rab 2441  df-v 2711  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-br 3962  df-opab 4022  df-xp 4585  df-cnv 4587  df-iota 5128  df-fv 5171  df-ov 5817  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-inn 8813 This theorem is referenced by:  nngt1ne1  8847  resqrexlemglsq  10899
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