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Theorem nn1gt1 8427
Description: A positive integer is either one or greater than one. This is for ; 0elnn 4422 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 2094 . . 3 (𝑥 = 1 → (𝑥 = 1 ↔ 1 = 1))
2 breq2 3841 . . 3 (𝑥 = 1 → (1 < 𝑥 ↔ 1 < 1))
31, 2orbi12d 742 . 2 (𝑥 = 1 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (1 = 1 ∨ 1 < 1)))
4 eqeq1 2094 . . 3 (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1))
5 breq2 3841 . . 3 (𝑥 = 𝑦 → (1 < 𝑥 ↔ 1 < 𝑦))
64, 5orbi12d 742 . 2 (𝑥 = 𝑦 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝑦 = 1 ∨ 1 < 𝑦)))
7 eqeq1 2094 . . 3 (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1))
8 breq2 3841 . . 3 (𝑥 = (𝑦 + 1) → (1 < 𝑥 ↔ 1 < (𝑦 + 1)))
97, 8orbi12d 742 . 2 (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1))))
10 eqeq1 2094 . . 3 (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1))
11 breq2 3841 . . 3 (𝑥 = 𝐴 → (1 < 𝑥 ↔ 1 < 𝐴))
1210, 11orbi12d 742 . 2 (𝑥 = 𝐴 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝐴 = 1 ∨ 1 < 𝐴)))
13 eqid 2088 . . 3 1 = 1
1413orci 685 . 2 (1 = 1 ∨ 1 < 1)
15 nngt0 8419 . . . . 5 (𝑦 ∈ ℕ → 0 < 𝑦)
16 nnre 8401 . . . . . 6 (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
17 1re 7466 . . . . . 6 1 ∈ ℝ
18 ltaddpos2 7910 . . . . . 6 ((𝑦 ∈ ℝ ∧ 1 ∈ ℝ) → (0 < 𝑦 ↔ 1 < (𝑦 + 1)))
1916, 17, 18sylancl 404 . . . . 5 (𝑦 ∈ ℕ → (0 < 𝑦 ↔ 1 < (𝑦 + 1)))
2015, 19mpbid 145 . . . 4 (𝑦 ∈ ℕ → 1 < (𝑦 + 1))
2120olcd 688 . . 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 8410 1 (𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wo 664   = wceq 1289  wcel 1438   class class class wbr 3837  (class class class)co 5634  cr 7328  0cc0 7329  1c1 7330   + caddc 7332   < clt 7501  cn 8394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-xp 4434  df-cnv 4436  df-iota 4967  df-fv 5010  df-ov 5637  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-inn 8395
This theorem is referenced by:  nngt1ne1  8428  resqrexlemglsq  10420
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