Theorem nn1gt1 9155

Description: A positive integer is either one or greater than one. This is for ℕ; 0elnn 4711 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.

Step	Hyp	Ref	Expression
1		eqeq1 2236	. . 3 ⊢ (𝑥 = 1 → (𝑥 = 1 ↔ 1 = 1))
2		breq2 4087	. . 3 ⊢ (𝑥 = 1 → (1 < 𝑥 ↔ 1 < 1))
3	1, 2	orbi12d 798	. 2 ⊢ (𝑥 = 1 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (1 = 1 ∨ 1 < 1)))
4		eqeq1 2236	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1))
5		breq2 4087	. . 3 ⊢ (𝑥 = 𝑦 → (1 < 𝑥 ↔ 1 < 𝑦))
6	4, 5	orbi12d 798	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝑦 = 1 ∨ 1 < 𝑦)))
7		eqeq1 2236	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1))
8		breq2 4087	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (1 < 𝑥 ↔ 1 < (𝑦 + 1)))
9	7, 8	orbi12d 798	. 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1))))
10		eqeq1 2236	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1))
11		breq2 4087	. . 3 ⊢ (𝑥 = 𝐴 → (1 < 𝑥 ↔ 1 < 𝐴))
12	10, 11	orbi12d 798	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝐴 = 1 ∨ 1 < 𝐴)))
13		eqid 2229	. . 3 ⊢ 1 = 1
14	13	orci 736	. 2 ⊢ (1 = 1 ∨ 1 < 1)
15		nngt0 9146	. . . . 5 ⊢ (𝑦 ∈ ℕ → 0 < 𝑦)
16		nnre 9128	. . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
17		1re 8156	. . . . . 6 ⊢ 1 ∈ ℝ
18		ltaddpos2 8611	. . . . . 6 ⊢ ((𝑦 ∈ ℝ ∧ 1 ∈ ℝ) → (0 < 𝑦 ↔ 1 < (𝑦 + 1)))
19	16, 17, 18	sylancl 413	. . . . 5 ⊢ (𝑦 ∈ ℕ → (0 < 𝑦 ↔ 1 < (𝑦 + 1)))
20	15, 19	mpbid 147	. . . 4 ⊢ (𝑦 ∈ ℕ → 1 < (𝑦 + 1))
21	20	olcd 739	. . 3 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1)))
22	21	a1d 22	. 2 ⊢ (𝑦 ∈ ℕ → ((𝑦 = 1 ∨ 1 < 𝑦) → ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1))))
23	3, 6, 9, 12, 14, 22	nnind 9137	1 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴))

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 713   = wceq 1395   ∈ wcel 2200   class class class wbr 4083  (class class class)co 6007  ℝcr 8009  0cc0 8010  1c1 8011   + caddc 8013   < clt 8192  ℕcn 9121

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-iota 5278  df-fv 5326  df-ov 6010  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-inn 9122

This theorem is referenced by:  nngt1ne1  9156  resqrexlemglsq  11549