Theorem nn1gt1 9041

Description: A positive integer is either one or greater than one. This is for ℕ; 0elnn 4656 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 2203	. . 3 ⊢ (𝑥 = 1 → (𝑥 = 1 ↔ 1 = 1))
2		breq2 4038	. . 3 ⊢ (𝑥 = 1 → (1 < 𝑥 ↔ 1 < 1))
3	1, 2	orbi12d 794	. 2 ⊢ (𝑥 = 1 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (1 = 1 ∨ 1 < 1)))
4		eqeq1 2203	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1))
5		breq2 4038	. . 3 ⊢ (𝑥 = 𝑦 → (1 < 𝑥 ↔ 1 < 𝑦))
6	4, 5	orbi12d 794	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝑦 = 1 ∨ 1 < 𝑦)))
7		eqeq1 2203	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1))
8		breq2 4038	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (1 < 𝑥 ↔ 1 < (𝑦 + 1)))
9	7, 8	orbi12d 794	. 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ ((𝑦 + 1) = 1 ∨ 1 < (𝑦 + 1))))
10		eqeq1 2203	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1))
11		breq2 4038	. . 3 ⊢ (𝑥 = 𝐴 → (1 < 𝑥 ↔ 1 < 𝐴))
12	10, 11	orbi12d 794	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 1 ∨ 1 < 𝑥) ↔ (𝐴 = 1 ∨ 1 < 𝐴)))
13		eqid 2196	. . 3 ⊢ 1 = 1
14	13	orci 732	. 2 ⊢ (1 = 1 ∨ 1 < 1)
15		nngt0 9032	. . . . 5 ⊢ (𝑦 ∈ ℕ → 0 < 𝑦)
16		nnre 9014	. . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
17		1re 8042	. . . . . 6 ⊢ 1 ∈ ℝ
18		ltaddpos2 8497	. . . . . 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 735	. . 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 9023	1 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴))

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 709   = wceq 1364   ∈ wcel 2167   class class class wbr 4034  (class class class)co 5925  ℝcr 7895  0cc0 7896  1c1 7897   + caddc 7899   < clt 8078  ℕcn 9007

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-xp 4670  df-cnv 4672  df-iota 5220  df-fv 5267  df-ov 5928  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-inn 9008

This theorem is referenced by:  nngt1ne1  9042  resqrexlemglsq  11204