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Theorem nn1gt1 8939
Description: A positive integer is either one or greater than one. This is for ; 0elnn 4615 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 2184 . . 3 (𝑥 = 1 → (𝑥 = 1 ↔ 1 = 1))
2 breq2 4004 . . 3 (𝑥 = 1 → (1 < 𝑥 ↔ 1 < 1))
31, 2orbi12d 793 . 2 (𝑥 = 1 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (1 = 1 ∨ 1 < 1)))
4 eqeq1 2184 . . 3 (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1))
5 breq2 4004 . . 3 (𝑥 = 𝑦 → (1 < 𝑥 ↔ 1 < 𝑦))
64, 5orbi12d 793 . 2 (𝑥 = 𝑦 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝑦 = 1 ∨ 1 < 𝑦)))
7 eqeq1 2184 . . 3 (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1))
8 breq2 4004 . . 3 (𝑥 = (𝑦 + 1) → (1 < 𝑥 ↔ 1 < (𝑦 + 1)))
97, 8orbi12d 793 . 2 (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1))))
10 eqeq1 2184 . . 3 (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1))
11 breq2 4004 . . 3 (𝑥 = 𝐴 → (1 < 𝑥 ↔ 1 < 𝐴))
1210, 11orbi12d 793 . 2 (𝑥 = 𝐴 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝐴 = 1 ∨ 1 < 𝐴)))
13 eqid 2177 . . 3 1 = 1
1413orci 731 . 2 (1 = 1 ∨ 1 < 1)
15 nngt0 8930 . . . . 5 (𝑦 ∈ ℕ → 0 < 𝑦)
16 nnre 8912 . . . . . 6 (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
17 1re 7944 . . . . . 6 1 ∈ ℝ
18 ltaddpos2 8397 . . . . . 6 ((𝑦 ∈ ℝ ∧ 1 ∈ ℝ) → (0 < 𝑦 ↔ 1 < (𝑦 + 1)))
1916, 17, 18sylancl 413 . . . . 5 (𝑦 ∈ ℕ → (0 < 𝑦 ↔ 1 < (𝑦 + 1)))
2015, 19mpbid 147 . . . 4 (𝑦 ∈ ℕ → 1 < (𝑦 + 1))
2120olcd 734 . . 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 8921 1 (𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wo 708   = wceq 1353  wcel 2148   class class class wbr 4000  (class class class)co 5869  cr 7798  0cc0 7799  1c1 7800   + caddc 7802   < clt 7979  cn 8905
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7890  ax-resscn 7891  ax-1cn 7892  ax-1re 7893  ax-icn 7894  ax-addcl 7895  ax-addrcl 7896  ax-mulcl 7897  ax-addcom 7899  ax-addass 7901  ax-i2m1 7904  ax-0lt1 7905  ax-0id 7907  ax-rnegex 7908  ax-pre-ltirr 7911  ax-pre-ltwlin 7912  ax-pre-lttrn 7913  ax-pre-ltadd 7915
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-xp 4629  df-cnv 4631  df-iota 5174  df-fv 5220  df-ov 5872  df-pnf 7981  df-mnf 7982  df-xr 7983  df-ltxr 7984  df-le 7985  df-inn 8906
This theorem is referenced by:  nngt1ne1  8940  resqrexlemglsq  11012
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