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Theorem nn1gt1 8714
Description: A positive integer is either one or greater than one. This is for ; 0elnn 4500 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 2122 . . 3 (𝑥 = 1 → (𝑥 = 1 ↔ 1 = 1))
2 breq2 3901 . . 3 (𝑥 = 1 → (1 < 𝑥 ↔ 1 < 1))
31, 2orbi12d 765 . 2 (𝑥 = 1 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (1 = 1 ∨ 1 < 1)))
4 eqeq1 2122 . . 3 (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1))
5 breq2 3901 . . 3 (𝑥 = 𝑦 → (1 < 𝑥 ↔ 1 < 𝑦))
64, 5orbi12d 765 . 2 (𝑥 = 𝑦 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝑦 = 1 ∨ 1 < 𝑦)))
7 eqeq1 2122 . . 3 (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1))
8 breq2 3901 . . 3 (𝑥 = (𝑦 + 1) → (1 < 𝑥 ↔ 1 < (𝑦 + 1)))
97, 8orbi12d 765 . 2 (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1))))
10 eqeq1 2122 . . 3 (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1))
11 breq2 3901 . . 3 (𝑥 = 𝐴 → (1 < 𝑥 ↔ 1 < 𝐴))
1210, 11orbi12d 765 . 2 (𝑥 = 𝐴 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝐴 = 1 ∨ 1 < 𝐴)))
13 eqid 2115 . . 3 1 = 1
1413orci 703 . 2 (1 = 1 ∨ 1 < 1)
15 nngt0 8705 . . . . 5 (𝑦 ∈ ℕ → 0 < 𝑦)
16 nnre 8687 . . . . . 6 (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
17 1re 7729 . . . . . 6 1 ∈ ℝ
18 ltaddpos2 8179 . . . . . 6 ((𝑦 ∈ ℝ ∧ 1 ∈ ℝ) → (0 < 𝑦 ↔ 1 < (𝑦 + 1)))
1916, 17, 18sylancl 407 . . . . 5 (𝑦 ∈ ℕ → (0 < 𝑦 ↔ 1 < (𝑦 + 1)))
2015, 19mpbid 146 . . . 4 (𝑦 ∈ ℕ → 1 < (𝑦 + 1))
2120olcd 706 . . 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 8696 1 (𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 680   = wceq 1314  wcel 1463   class class class wbr 3897  (class class class)co 5740  cr 7583  0cc0 7584  1c1 7585   + caddc 7587   < clt 7764  cn 8680
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-xp 4513  df-cnv 4515  df-iota 5056  df-fv 5099  df-ov 5743  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-inn 8681
This theorem is referenced by:  nngt1ne1  8715  resqrexlemglsq  10745
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